# Logarithmic Equations

We can use the laws of logarithms to write equations in a different form. This can be particularly useful if an unknown appears as an index (exponent). $$2^x=7$$ For the logarithmic function, for every value of $y$, there is only one corresponding value of $x$. $$y=5^x$$ We can therefore take the logarithm of both sides of an equation without changing the solution. However, we can only do this if both sides are positive.

The equation $\log_{a}{y} = x$ is an example of a general logarithmic equation. Laws of logarithms and exponents (indices) are used to solve these equations.

### Example 1

Write $y=a^3b^2$ as logarithmic equations in base 10.

### Example 2

Write $\log{x}=\log{a} + 2\log{b} - 3\log{c}$ without logarithms.

### Example 3

Write $2\log_{5}{x}=\log_{5}{3a} + 2$ without logarithms.

### Example 4

Write $x$ in terms of $y$ given $y=5 \times 2^x$.

### Example 5

Find $x$ if $\log_{3}{9} = x-2$.

### Example 6

Find $x$ if $\log_{4}{x}=-2$.

### Example 7

Find $x$ if $3\log_{x}{16}=6$, $x>0$.

### Example 8

Solve $\log_{x}{\dfrac{1}{125}} = -3$.

### Example 9

Solve $\log_{10}{x} + \log_{10}{(x-3)} = \log_{10}{4}$.