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# Solve the equation $\log_2(\frac{8}{2^x}-1)=x-2$

This equation comes from the book of Prilepko. It looks sufficiently simple but I want to know if my solution is correct.

$\log_2(\frac{8}{2^x}-1)=x-2$

I have only been able to transform into this:

$\log_2(\frac{8-2^x}{2^x})=x-2$

$\iff$ $\log_2(8-2^x)=2x-2$

$\iff$ $8-2^x=2^{2x-2}$

$\iff$ $32-4.2^x-2^{2x}= 0$

Let $t=2^x$

$-t^2-4t+32=0$

This equation will produce $t=-8$ and $t=4$

Therefore $x=2$.

Another question that I wish to ask is when solving these logarithmic equations, one very often substitute a dummy variable to overcome the restriction caused by the order of operations. For example, you cannot do much directly with $5^{2x}-130.5^x+625$. But let $t=5^x$, then you can easily factorize the this equation into $(t-125)(t-5)=0$, hence $(5^x-125)(5^x-5)=0$. Isn't this a sort of manipulating symbols according to certain prescribed rules? Is there any logical difficulty with this kind of operation? Since by substituting a new variable, it means that this variable must inherit all properties which the object of the substitution originally possesses. For example, if $a^x$ is the object of substitution for $t$, then if $a<0$ and x can assumed a form of rational power, doesn't this make no sense at all?

Do you know any example in elementary mathematics where this method of substitution can yield contradictory result?

• Your solution looks correct for the first part, and I believe that is the nicest solution. – Carl Schildkraut Mar 15 '18 at 19:48
• there is just a typo $$\log_2\left(\frac{8-2^x}{\color{red}{2^x}}\right)=x-2$$ – gimusi Mar 15 '18 at 19:48
• Thanks, I have corrected it. But is this method of substitution justified logically? Is there circumstances where you apply substitution and it yields a contradictory result? – James Warthington Mar 15 '18 at 19:51
• Sometimes we get extraneous solutions... solutions to an equation we got from the original equation but sometimes their domains differ. – randomgirl Mar 15 '18 at 19:56
• If $x$ is a real number then $2^x$ is a positive real number. Making the substitution that $t = 2^x$ then $t$ must be a positive real. And if the algebra showed a solution that worked where t was a negative real, it is still does not a solution for $x.$ And if your algebra lead you to the conclusion that solutions only existed for negative values of $t.$ Then there is no solution for real values of $x.$ What is all of this talk of "real numbers" there are numbers that are not real... complex numbers.. for which $2^z$ may be negative. – Doug M Mar 15 '18 at 19:58

Your solution is fine, well done. Note that you can omit the second step by doing $$\log_2\left(\frac{8}{2^x}-1\right)=x-2\implies8\cdot2^{-x}-1=2^{x-2}\implies8-2^x=2^{2x-2}$$

A substitution of the form $t=a^x$ for (positive) $a$ can be used at any time without doubting whether it would yield unwanted solutions, since if $t$ is negative then it can be automatically rejected.

• How does this answer the question? – user535339 Mar 15 '18 at 19:52
• Well thanks for checking my answer. But do you know any kind of problems which by substituting a dummy variable, you may get a result that doesn't make sense? – James Warthington Mar 15 '18 at 19:53
• @idk The OP asks, "I want to know if my solution is correct." TheSimpliFire answers, "Your solution is fine." – saulspatz Mar 15 '18 at 19:55
• @saulspatz That was my mistake, I didn't read the first part of the question. – user535339 Mar 15 '18 at 19:57
• Thank you very much! I have no other questions. – James Warthington Mar 15 '18 at 20:07

First of all, yes your solution is correct and you can also check it by putting it again in given equation. And secondly, about your second question, there is nothing wrong in replacing a value by a variable and no ambiguity will be created in this process. You can replace it with even more or less complex variable and it will remain to its own properties.

Your answer $x=2$ is correct. You have made minor typo's which you can fix.

I let y=2^x and substituted in $$2^{2x-2}=8-2x$$ to get $$y^2 +4y-32=0$$

Solving for $y$ and we get $y=4$ or $y=-8$

Clearly, $y=-8$ is not acceptable.

Thu $y=4$ which implies $x=2$

If you want "to stretch" the logarithm method or solution to the very end here it is:

$$\log_{2} \left ( \frac{8}{2^{x}} - 1 \right ) = x-2$$

$$\log_{2} \left ( \frac{8-2^{x}}{2^{x}} \right ) = x-2$$

$$\log_{2} \left (8-2^{x} \right ) - \log_{2} \left ( 2^{x} \right ) = x-2$$

$$\log_{2} \left (2^{3}-2^{x} \right ) - x = x-2$$

$$\log_{2} \left (2^{3}-2^{x} \right ) = 2x-2$$

$$\log_{2} \left (2^{3} \left ( 1- 2^{x-3}\right ) \right ) = 2x-2$$

$$\log_{2} \left (2^{3}\right ) + \log_{2} \left ( 1- 2^{x-3} \right ) = 2x-2$$

$$3\log_{2} \left (2\right ) + \log_{2} \left ( 1- 2^{x-3} \right ) = 2x-2$$

$$3 + \log_{2} \left ( 1- 2^{x-3} \right ) = 2x-2$$

$$\log_{2} \left ( 1- 2^{x-3} \right ) = 2x-5$$

Using the $\textrm{Antilogarithm}$ (see here or here for any details)

$$\textrm{antilog}_{2} \left ( \log_{2} \left ( 1- 2^{x-3} \right ) \right ) = \textrm{antilog}_{2} \left ( 2x-5 \right )$$

$$1- 2^{x-3}= 2^{2x-5}$$

Rearranging the above equation translates into:

$$1- 2^{x}\times 2^{-3}= 2^{2x}\times 2^{-5}$$

$$2^{2x}\times 2^{-5} - 2^{x}\times 2^{-3} - 1 = 0$$

Multiplying by $2^{5}$ becomes into the expression you had found:

$$2^{2x} - 2^{x}\times 2^{2} - 32 = 0$$

From then on it is consistent with a quadratic equation:

$$A=2^{x}$$

$$A_{1,2}=\frac{-4\pm \sqrt{16+128}}{2}=\frac{-4\pm \sqrt{144}}{2}$$

$$A_{1,2}=\frac{-4\pm 12}{2}\,,A_{1}=\frac{-4+12}{2}=4\,,A_{2}=\frac{-4-12}{2}=-8$$

Since we cannot obtain negative results from a power the second solution is discarded. Therefore:

$$2^{x}=4$$

$$x=2$$

Becoming the result you had just found. So it is correct. As mentioned this is more or less the same of what others before me had just commented.

observe that $8=2^3$.

$$\ln (2^{3-x}-1)=(x-2)\ln (2)=\ln (2^{x-2})$$

$$2^{3-x}-2^{x-2}=1$$

Put $2^x=t >0$.

it becomes $$8/t-t/4-1=0$$

$$t^2+4t-32=0$$ $t=4 \implies x=2$

It is correct and the best method to solve this kind of equations, fix just a typo

$$\log_2\left(\frac{8-2^x}{\color{red}{2^x}}\right)=x-2$$

Regarding to the general question note that the substitution $a^x=t$ is nice since we are not limiting the possible values for $a^x>0$. Of course we can obtain some not positive solution but this is not a problem since we can simply discard them as you have done in this example. Otherwise it should be uncorrect substitute a variable $t$ which can assume also negative values with $a^x$ because in this way we could loss solutions.

• How does this answer the question? – user535339 Mar 15 '18 at 19:46
• ah ok I didn't recognize that it was just a typo, sorry! – gimusi Mar 15 '18 at 19:47


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text:
Q

# Form the pair of linear equations for the following problems and find their solution by substitution method.  (iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge

Q3.    Form the pair of linear equations for the following problems and find their solution by substitution method.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

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Let the fixed charge is x and the per km charge is y.

Now According to the question

$x+10y=105.......(1)$

And

$x+15y=155.......(2)$

Now, From (1) we have,

$x=105-10y........(3)$

Substituting this value of x in (2), we have

$105-10y+15y=155$

$\Rightarrow 5y=155-105$

$\Rightarrow 5y=50$

$\Rightarrow y=10$

Now, Substituting this value in (3)

$x=105-10y=105-10(10)=105-100=5$

Hence, the fixed charge is 5 Rs and the per km charge is 10 Rs.

Now, Fair For 25 km :

$\Rightarrow x+25y=5+25(10)=5+250=255$

Hence fair for 25km is 255 Rs.

Exams
Articles
Questions


text:
# Source

print, format, and %, evolved.

Q: It's been forty years since C introduced printf() and the basic formatted printing of positional parameters. Isn't it time for an upgrade?

A: Yes! ZOMG, yes!

say supplements or replaces Python's print statement/function, format function/method, and % string interpolation operator with higher-level facilities:

• Straightforward string formatting with DRY, Pythonic templates that piggyback the built in format() method, formatting syntax, and well-proven underlying engine.
• A single output mechanism compatible with both Python 2.x and Python 3.x.
• Indentation and wrapping (to help stucture output)
• Convenience printing functions for horizontal rules (lines), titles, and vertical whitespace.

## Usage

from say import say, fmt, show

x = 12
nums = list(range(4))

say("There are {x} things.")
say("Nums has {len(nums)} items: {nums}")

yields:

There are 12 things.
Nums has 4 items: [0, 1, 2, 3]

say is basically a simpler, nicer recasting of:

print "There are {} things.".format(x)
print "Nums has {} items: {}".format(len(nums), nums)

(NB in Python 2.6 one must number each of the {} placeholders--e.g. "Nums has {0} items: {1}"-- in order to avoid a ValueError: zero length field name in format error. Python 2.7 and later assume the placeholders are sequential.)

The more items that are being printed, and the complicated the format invocation, the more valuable having it stated in-line becomes. Note that full expressions are are supported. They are evaluated in the context of the caller.

## Printing Where You Like

say() writes to a list of files--by default just sys.stdout. But with it simple configuration call, it will write to different--even multiple--files:

from say import say, stdout

say.setfiles([stdout, "report.txt"])
say(...)   # now prints to both stdout and report.txt

This has the advantage of allowing you to both capture and see program output, without changing any code (other than the config statement). You can also define your own targeted Say instances:

from say import say, Say, stderr

err = say.clone().setfiles([stderr, 'error.txt'])
err("Failed with error {errcode}")  # writes to stderr, error.txt

Note that stdout and stderr are just convenience aliases to the respective sys equivalents.

## Printing When You Like

If you want to stop printing for a while:

say.set(silent=True)  # no printing until set to False

Or transiently:

say(...stuff..., silent=not verbose) # prints iff bool(verbose) is True

Of course, you don't have to print to any file. There's a predefined sayer fmt() that works exactly like say() and inherits most of its options, but doesn't print. (The C analogy: say : fmt :: printf : sprintf.)

## Indentation and Wrapping

Indentation is a common way to display data hierarchically. say will help you manage it. For example:

say('TITLE')
for item in items:
say(item, indent=1)

will indent the items by one indentation level (by default, each indent level is four spaces, but you can change that with the indent_str option).

If you want to change the default indentation level:

say.set(indent=1)      # to an absolute level
say.set(indent='+1')   # strings => set relative to current level

...

say.set(indent=0)      # to get back to the default, no indent

Or you can use a with construct:

with say.settings(indent='+1'):
say(...)

# anything say() emits here will be auto-indented +1 levels

# anything say() emits here, after the with, will not be indented +1

And if you have a lot of data or text to print, you can easily wrap it:

say("This is a really long...blah blah blah", wrap=40)

Will automatically wrap the text to the given width (using Python's standard textwrap module).

While it's easy enough for any print statement or function to have a few space characters added to its format string, it's easy to mistakenly type too many or too few spaces, or to forget to type them in some format strings. And if you're indenting strings that themselves may contain multiple lines, the simple print approach breaks because won't take multi-line strings into account. And it won't be integrated with wrapping.

say, however, simply handles the indent level and wrapping, and it properly handles the multi-line string case. Subsequent lines will be just as nicely and correctly indented as the first one--something not otherwise easily accomplished without adding gunky, complexifying string manipulation code to every place in your program that prints strings.

This starts to illustrate the "do the right thing" philosophy behind say. So many languages' printing and formatting functions a restricted to "outputting values" at a low level. They may format basic data types, but they don't provide straightforward ways to do neat text transformations like indentation that let programmers rapidly provide correct, highly-formatted ouput. Over time, say will provide higher-level formatting options. For now: indentation and wrapping.

## Encodings

say() and fmt() try to work with Unicode strings, for example providing them as return values. But character encodings remain a fractious and often exasperating part of IT. When writing formatted strings, say handles this by encoding into utf-8.

If you are using strings containing utf-8 rather than Unicode characters, say may complain. But it complains in the same places the built-in format() does, so no harm done. (Python 3 doesn't generally allow utf-8 in strings, so it's cleaner on this front.)

You can get creative with the encoding:

say('I am a truck!', encoding='base64')  # SSBhbSBhIHRydWNrIQo=

Or change the default:

say.set(encoding='rot-13')

Knock yourself out with all the exciting opportunites! If you really want the formatted text returned just as it is written to files, use the encoded option. Set to True and it returns text in the output encoding. Or set to an actual encoding name, and that will be the return encoding.

say() returns the formatted text with one small tweak: it removes the final newline if a newline is the very last character. Though odd, this is exactly what you need if you're going to print or say the resulting text without a gratuitous "extra" newline.

## Titles and Horizontal Rules

say defines a few convenience formatting functions:

say.title('Errors', sep='-')
for i,e in enumerate(errors, start=1):
say("{i:3}: {e['name'].upper()}")

might yield:

--------------- Errors ---------------
1: I/O ERROR
2: COMPUTE ERROR

A similar method hr produces just a horizontal line, like the HTML <hr> element. For either, one can optionally specify the width (width), character repeated to make the line (sep), and vertical separation/whitespace above and below the item (vsep). Good options for the separator might be be '-', '=', or parts of the Unicode box drawing character set.

## Python 3

Say works virtually the same way in Python 2 and Python 3. This can simplify software that should work across the versions, without all the from __future__ import print_function hassle.

say attempts to mask some of the quirky compexities of the 2-to-3 divide, such as string encodings and codec use.

## Alternatives

• ScopeFormatter provides variable interpolation into strings. It is amazingly compact and elegant. Sadly, it only interpolates Python names, not full expressions. say has full expressions, as well as a framework for higher-level printing features beyond ScopeFormatter's...um...scope.

• Even simpler are invocations of % or format() using locals(). E.g.:

name = "Joe"
print "Hello, %(name)!" % locals()
# or
print "Hello, {name}!".format(**locals())

Unfortunately this has even more limitations than ScopeFormatter: it only supports local variables, not globals or expressions. And the interpolation code seems gratuitous. Simpler:

say("Hello, {name}!")

## Notes

• The say name was inspired by Perl's say, but the similarity stops there.
• The show debug printing functions have been split into a separate package, show.
• Automated multi-version testing is managed with the wonderful pytest and tox. say is now successfully packaged for, and tested against, all late-model verions of Python: 2.6, 2.7, 3.2, and 3.3, as well as PyPy 1.9 (based on 2.7.2).
• Debug printing via show() is now operational. Draft support for interactive Python and iPython. Works within imported modules, and at the interactive prompt, but cannot be used within interactively defined functions. Unknown whether that is a hard limit of Python introspection, or something that can be worked around over time.
• say has greater ambitions than just simple template printing. It's part of a larger rethinking of how output should be formatted. show() is an initial down-payment. Stay tuned for more.
• In addition to being a practical module in its own right, say is testbed for options, a package that provides high-flexibility option, configuration, and parameter management.

## To-Dos

• Provide code that allows pylint to see that variables used inside the say and fmt format strings are indeed thereby used.

## Installation

pip install say

To easy_install under a specific Python version (3.3 in this example):

python3.3 -m easy_install say

(You may need to prefix these with "sudo " to authorize installation.)