braffter92

2022-10-03

Finding values of x for logarithm

The question is to find the numbers of x which satisfy the equation.

$${\mathrm{log}}_{x}10={\mathrm{log}}_{4}100.$$

I have

$$\begin{array}{rl}\frac{\mathrm{ln}10}{\mathrm{ln}x}& =\frac{\mathrm{ln}100}{\mathrm{ln}4}\\ \frac{\mathrm{ln}10}{\mathrm{ln}x}& =\frac{2\mathrm{ln}10}{2\mathrm{ln}2}\end{array}$$

What would I do after this step?

The question is to find the numbers of x which satisfy the equation.

$${\mathrm{log}}_{x}10={\mathrm{log}}_{4}100.$$

I have

$$\begin{array}{rl}\frac{\mathrm{ln}10}{\mathrm{ln}x}& =\frac{\mathrm{ln}100}{\mathrm{ln}4}\\ \frac{\mathrm{ln}10}{\mathrm{ln}x}& =\frac{2\mathrm{ln}10}{2\mathrm{ln}2}\end{array}$$

What would I do after this step?

Kaleb Harrell

Beginner2022-10-04Added 14 answers

All right. First multiply by $ln(x)$ and by $ln(2)$. You get

$ln(10)ln(2)=ln(10)ln(x)$

Now divide by $ln(10)$. This gives you

$ln(2)=ln(x)$

Now you apply the exponential function on both sides to get rid of the logarithm:

$\underset{=x}{\underset{\u23df}{{e}^{ln(x)}}}=\underset{=2}{\underset{\u23df}{{e}^{ln(2)}}}$

So $x=2$

$ln(10)ln(2)=ln(10)ln(x)$

Now divide by $ln(10)$. This gives you

$ln(2)=ln(x)$

Now you apply the exponential function on both sides to get rid of the logarithm:

$\underset{=x}{\underset{\u23df}{{e}^{ln(x)}}}=\underset{=2}{\underset{\u23df}{{e}^{ln(2)}}}$

So $x=2$

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