# Thread:

1. ## Determing range of values of x & y

Hi

If real variables x and y are related by the equation:

ln(3-y) = 3ln(2x+1)-x

then how do i:

a). determine the range of values of x and y for which expressions on each side of equation are defined, and

b). find y explicitly as a function of x, that is, express the equation in the form y=f(x)

Thanks

2. Originally Posted by moolimanj
Hi

If real variables x and y are related by the equation:

ln(3-y) = 3ln(2x+1)-x

then how do i:

a). determine the range of values of x and y for which expressions on each side of equation are defined, and

b). find y explicitly as a function of x, that is, express the equation in the form y=f(x)

Thanks
to a) The ln-function is defined for positive arguments:

$3-y>0~\implies~y<3$ and

$2x+1>0~\implies~x>-\frac12$

To b)

$\ln(3-y) = 3\ln(2x+1)-x~\implies~3-y=\frac{(2x+1)^3}{e^x}~\implies~y=3- \frac{(2x+1)^3}{e^x}~,~x>-\frac12$

3. a) The conditions are
$3-y>0$
$2x+1>0$
Now, can you solve them?

b) The equation can be written as
$\displaystyle\ln(3-y)=\ln(2x+1)^3-\ln e^x\Leftrightarrow\ln(3-y)=\ln\frac{(2x+1)^3}{e^x}\Leftrightarrow$
$\displaystyle\Leftrightarrow 3-y=\frac{(2x+1)^3}{e^x}\Leftrightarrow y=3-\frac{(2x+1)^3}{e^x}$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.7817173004150391, "perplexity_flag": "middle"}