Question Video: Solving Logarithmic Equations Mathematics

Video Transcript

Find 𝑥 such that log of four 𝑥 minus four is equal to two.

Let’s write down the equation again. What is the base of this logarithm? When there isn’t a small subscript number next to the log, it’s a logarithm base 10.

We can write in this base explicitly if we want to. But you should always remember that a log without a base written explicitly means a log base 10.

Recall that the equation log base 𝑎 of 𝑏 equals 𝑐, which is in logarithmic form, is equivalent to the equation 𝑎 to the power of 𝑐 equals 𝑏, which is in exponential form. Let’s rewrite the equation we have to solve in exponential form.

Comparing to the general rule we see that 𝑎 is 10. And so our equation is going to be 10 to the power of 𝑐 equals 𝑏. 𝑏 is four 𝑥 minus four. So we have that 10 to the power of 𝑐 equals four 𝑥 minus four. And finally 𝑐 is two. So we have 10 to the power of two is four 𝑥 minus four.

We give our focus to the equation in exponential form. We know what’s 10 to the power of two is. It’s 100. And so we have the equation 100 equal four 𝑥 minus four. This is now a linear equation that we know how to solve.

We add four to both sides and then divide by four to find that 26 equals 𝑥. And so 𝑥 equals 26. Our answer, therefore, the value of 𝑥 such that the log of four 𝑥 minus four is equal to two is 26.