Question Video: Solving Logarithmic Equations With A Single ...

Video Transcript

Solve for 𝑥. log base two of 𝑥 plus one equals two-thirds.

Since we are given a logarithmic equation with a single log, we can rearrange the equation to make the argument, 𝑥 plus one, the subject by rewriting it as an exponential equation. We begin by recalling the definition of a logarithm. For an exponential equation in the form 𝑎 to the power of 𝑥 equals 𝑛, this can be written as the logarithmic equation log base 𝑎 of 𝑛 equals 𝑥. This definition can be used to solve this logarithmic equation by rewriting it in its exponential form.

Using the definition of a logarithm, we see that two-thirds is our exponent and two is our base. Therefore, we rewrite log base two of 𝑥 plus one equals two-thirds as two to the power of two-thirds equals 𝑥 plus one. To finish solving for 𝑥, we subtract one from each side of our new equation. We are then left with 𝑥 equals two to the power of two-thirds minus one.

We want our answer to be as simplified as possible, so we will try to evaluate two to the power of two-thirds. We recall that a rational exponent is a way to write both an exponent and a root in a more condensed way. The denominator is the index of the root and the numerator is the integer e𝑥ponent. Therefore, two to the power of two-thirds equals the cube root of two squared, where the exponent can be either inside or outside the radical. But since neither two nor four are perfect cube numbers, there is no need to write our final answer in this form.

It is a good idea to perform a check of our answer at the end, especially because sometimes logarithmic equations have an extraneous solution. We go ahead and substitute our answer for 𝑥 in the left side of the original equation. We want to know if this will equal two-thirds. The expression inside the logarithm simplifies to two to the power of two-thirds. According to the power law of logarithms, log base two of two to the two-thirds power can be rewritten as two-thirds times log base two of two. And we know that log base two of two equals one. So, two-thirds times log base two of two equals two-thirds. This is confirmation of our final answer. 𝑥 equals two to the two-thirds power minus one.