finding $x$ when it is the power in both the numerator and denominator

Question

I have an equation and I've simplified it to this:

$$\frac {24}{19}=\frac {1.01583^{12x}}{1.01583^{12x}-1}$$

I'm having trouble finding $x$? I realize I have to find the log at some point, but am struggling as to when and where I do so?

Answer 1

let $$a=1.01583^{12x}$$ then we have to solve $$\frac{24}{19}=\frac{a}{a-1}$$ from here we get $$24a-24=19a$$ thus we have $$a=\frac{24}{5}$$ can you proceed? from here we get $$\ln(a)=12x\ln(1.01583)$$ thus $$x=\frac{\ln(a)}{12\ln(1.01583)}$$

Answer 2

Generalize the problem, solve $x$ when all the variables are real and positive:

$$\text{a}=\frac{\text{b}^{\text{c}x}}{\text{b}^{\text{c}x}-\text{d}}\Longleftrightarrow\text{a}=\frac{\frac{\text{b}^{\text{c}x}}{\text{b}^{\text{c}x}}}{\frac{\text{b}^{\text{c}x}}{\text{b}^{\text{c}x}}-\frac{\text{d}}{\text{b}^{\text{c}x}}}\Longleftrightarrow$$ $$\text{a}=\frac{1}{1-\frac{\text{d}}{\text{b}^{\text{c}x}}}\Longleftrightarrow\frac{1}{\text{a}}=1-\frac{\text{d}}{\text{b}^{\text{c}x}}\Longleftrightarrow$$ $$\frac{1}{\text{a}}-1=-\frac{\text{d}}{\text{b}^{\text{c}x}}\Longleftrightarrow1-\frac{1}{\text{a}}=\frac{\text{d}}{\text{b}^{\text{c}x}}\Longleftrightarrow$$ $$\frac{\text{d}}{1-\frac{1}{\text{a}}}=\text{b}^{\text{c}x}\Longleftrightarrow\frac{\log_{\text{b}}\left(\frac{\text{d}}{1-\frac{1}{\text{a}}}\right)}{\text{c}}=x$$

So, for your prolem we get:

$$x=\frac{\log_{1.01583}\left(\frac{1}{1-\frac{1}{\frac{24}{19}}}\right)}{12}=\frac{\log_{1.01583}\left(\frac{24}{5}\right)}{12}\approx8.322799606$$