 Online Calculators

# Logarithm Equation Calculator

Logarithm Equation Solver
$\log_{b}x = y$
Enter any two values or fractions
to solve for the third value

## Calculator Use

This calculator will solve the basic log equation logbx = y for any one of the variables as long as you enter the other two.

The logarithmic equation is solved using the logarithmic function:

$$x = \log_{b}b^x$$

which is equivalently

$$x = b^{log_{b}x}$$

## How to solve the logarithmic equation

If we have the equation used in the Logarithm Equation Calculator

$$\log_{b}x = y \;(1)$$

We can say the following is also true

$$b^{\log_{b}x} = b^{y} \;(2)$$

Using the logarithmic function where

$$x = b^{log_{b}x}$$

We can rewrite our equation (2) to solve for x

$$x = b^{y} \;(3)$$

Solving for b in equation (3) we have

$$b = \sqrt[y]{x}$$

Solving for y in equation (3)

$$x = b^{y} \;(3)$$

take the log of both sides:

$$\log_{10}x = \log_{10}b^y$$

Using logarithmic identity we rewrite the equation:

$$\log_{10}x = y \cdot \log_{10}b$$

Dividing both sides by log b:

$$y = \dfrac{\log_{10}x}{\log_{10}b} = \dfrac{\log_{}x}{\log_{}b}$$

Note that writing log without the subscript for the base it is assumed to be log base 10 as in log10.

### Example 1: Solve for y in the following logarithmic equation

If we have

$$\log_{3}5 = y$$

then it is also true that

$$3^{\log_{3}5} = 3^{y}$$

Using the logarithmic function we can rewrite the left side of the equation and we get

$$5 = 3^{y}$$

To solve for y, first take the log of both sides:

$$\log_{}5 = \log_{}3^y$$

By the identity log xy = y · log x we get:

$$\log_{}5 = y \cdot \log_{}3$$

Dividing both sides by log 3:

$$y = \dfrac{\log_{}5}{\log_{}3}$$

Using a calculator we can find that log 5 ≈ 0.69897 and log 3 ≈ 0.4771 2 then our equation becomes:

$$n = \dfrac{\log_{}5}{\log_{}3} = \dfrac{0.69897}{0.47712} = 1.46497$$

Therefore, putting y back into our original equation

$$\log_{3}5 = 1.46497$$

### Example 2: Solve for b in the following logarithmic equation

If we have

$$\log_{b}16 = 2$$

then it is also true that

$$b^{\log_{b}16} = b^{2}$$

Using the logarithmic function we can rewrite the left side of the equation and we get

$$16 = b^{2}$$

Solving for b by taking the 2nd root of both sides of the equation

$$b = \sqrt{16} = 4$$

Therefore, putting b back into our original equation

$$\log_{4}16 = 2$$

Cite this content, page or calculator as:

Furey, Edward "Logarithm Equation Calculator"; CalculatorSoup, https://www.calculatorsoup.com - Online Calculators 