Equations with logarithms

Exponential and logarithmic growth: Logarithmic growth

Equations with logarithms

The logarithm allows us to write down solutions to equations like #4^x = 5# with unknown #x#; here: #x=\log_4(5)#.

Conversely, equations in which the unknown #x# only occurs through #\log_4(x)# can be solved by exponentiation of the answer. For example, the solution to the equation #\log_4(x) = \frac{5}{2}# is #x=32#.

Below we give some examples.

Determine an exact expression for the value of #x# that satisfies #\log_{7}\left(x\right)=3#.

Give your answer in the form #x = \ldots#, where the dots are replaced by a number.

#x=343#

Using the definition of the logarithm we can rewrite the equation as #x={7}^{3}=343#.

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