Solving Log Equations

Introduction

Recall the rules of logs :

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$$\log_a(xy)=\log_ax+\log_ay$$ $$\log_a\left(x^q\right)=q\log_ax$$ $$\log_a\left(\tfrac{1}{x}\right)=-\log_ax$$ $$a^{\log_ax}=x$$ $$\log_a\left(\tfrac{x}{y}\right)=\log_ax-\log_ay$$ $$\log_a1=0$$ $$\log_a\left(a^x\right)=x$$

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In the same way we approach fractions, an equation with multiple logs can be reduced to an equation with only one log (in most cases).

Example

$e$ and the natural log

In mathematics, a special name is given to the constant $e$ (in the same way we give a special name to $\pi$). The value of $e$ is approximately 2.71828..., and is irrational. It is used to represent growth, which was discovered when studying compound interest in 1683.

The natural log is a log with a base of $e$. And is denoted by :

$$\ln(x)=\log_e(x)$$

Example