In Exercises 71–78, use common logarithms or natural logarithms a...

Question

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.

log5 13

Accepted Answer

Hello, everyone in this video, we're gonna be looking at this practice problem where we want to evaluate the expression log base seven of 31 and we can use our calculator. However, notice how we have a log base seven. And if you look at the calculator, you only see a log button. Well, most calculators assume log base 10 or the common logarithm. So we need to change our log base seven into log base 10 in order to evaluate using a calculator. And because we're changing the base, we can use the change of base property of logarithms to go from log base seven to log base 10. And the change of base property says that if we have log log base B of M, we can change it into log base A using a fraction. So notice how we went in with one log and we'll have two, but we'll have two logs with the base that we want. So we can evaluate them. And the fraction will be log base A of M and that's divided by log base A of B. So notice how the unwanted or not calculable base B ends up in the denominator of the new base. And this is what we're gonna use to help us evaluate our expression. So we have log base seven of 31. And if I change that into log base 10, I'll have log base 10 of 31 divided by log base 10 of seven. So now you can see that we have two logs, log base 10 of 31 and log base 10 of seven. And although it's a little bit more work, we can now evaluate them in the calculator because they both have base 10. So log base 10 of 31 is equal to 1.4913. And that's still being divided by log base 10 of seven, which is 0.8450 when we plug it into the calculator. And if I complete this division, I get that log base seven of 31 is approximately equal to 1.7646. And this is my final answer when I evaluate the expression. And you can see that, that aligns with answer choice B so hopefully this video helped and see you next time.

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In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log5 13