# Solving Logarithmic equations with Different bases Questions with solutions

In logarithms, solving the log equations with different bases is one type of logarithm questions. The examples on solving the logarithm equations with different bases practice questions worksheet with answers is given here for those who want to learn how to solve the logarithmic equations with different bases by the log rules and solutions to understand the process of solving the logarithmic equations with different bases by the logarithmic laws in mathematics.

## Arithmetic form

The log equations with different bases are formed by the arithmetic quantities in some cases as follows.

$(1).\,\,$ Solve $\log_{\Large \frac{1}{2}}{\Big(\log_{\large x}{\big(\log_{4}{(32)}\big)}\Big)}$ $\,=\,$ $2$

$(2).\,\,$ Solve $\log_{2}{x}$ $+$ $\log_{4}{x}$ $+$ $\log_{16}{x}$ $\,=\,$ $7$

$(3).\,\,$ Solve $\log_{\sqrt{2}}{\Big(\log_{\large 2}{\big(\log_{4}{(x-15)}\big)}\Big)}$ $\,=\,$ $0$

Let’s learn how to solve the logarithmic equations in arithmetic quantities with different bases by the logarithmic properties.

### Algebraic form

In some cases, the log equations with different bases are formed by the algebraic quantities as follows.

$(1).\,\,$ Solve $2\log_{\large x}{a}$ $+$ $\log_{\large ax}{a}$ $+$ $3\log_{\large a^2x}{a}$ $\,=\,$ $0$

$(2).\,\,$ Solve $\log_{\large e}{\log_{\large e}{\log_{\large e}{x}}}$ $\,=\,$ $0$

$(3).\,\,$ Solve $\log_{\large (2x+3)}{(6x^2+3x+21)}$ $\,=\,$ $4$ $-$ $\log_{\large (3x+7)}{(4x^2+12x+9)}$

Let us learn how to solve the logarithm equations in algebraic quantities with different bases by the logarithmic formulas.

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