Calculators

Algebra

Calculator Use

This calculator will solve the basic log equation log_bx = 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 x^y = 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

\[ b = \sqrt[2]{16} = 4 \]

Therefore, putting b back into our original equation

\[ \log_{4}16 = 2 \]