Logarithmic equations can be equivalently written using exponents. In general, we can say that the following two equations are equivalent:
Notice that the base of the logarithm is the base of the exponential expression. Additionally, the input of the logarithmic function is the output of the exponential function.
Rewriting Logarithmic Equations as Exponential Equations
For some logarithmic equations, it may be helpful to rewrite the equation as an equivalent exponential equation.
One strategy in solving logarithmic equations is to rewrite it as an exponential equation. In many cases, by doing so, the equation will have an exponential expression on one side of the equation, and an isolated variable on the other side. We can then evaluate the exponential expression to find the solution to the equation.
Using the Log Expression as an Exponent
Another method to solving log equations involves applying the inverse relationship between exponents and logs in a slightly different way that you may be used to. We can use the base of the logarithm as a base to an exponent, and place the logarithmic expression as an exponent in the equation. We'll have to do this to both sides of the equation.
This is essentially a more explicit explanation of the relationship between logarithmic equations and exponential equations. We can use the base of the log to create an exponential equation with the same base. Since logarithms and exponents are inverse operations, this undoes any log or exponent operation, leaving only the argument of the log on one side of the equation, and an exponential expression on the other.