Video Transcript
Calculate log 16 minus log 25 over log 64 minus log 125.
We note first that the logarithm without a specified base refers to the logarithm to the base 10. Each of the terms in our expression is therefore logarithm to the base 10 of some number. To evaluate the given expression, we’re going to begin by using our knowledge of powers of two and powers of five to simplify. We know that 16 is two raised to the power four and 64 is two raised to the power six. We know also that 25 is five squared and 125 is five raised to the power three. Hence, the arguments of our logarithms are either powers of two or powers of five. And writing out each of our terms, we have log of 16 is log of two raised to the power four. Log of 64 is log of two raised to the power six. The logarithm of 25 is the log of five raised to the power two. And the logarithm of 125 is the log of five raised to the power three.
And rewriting our expression, we then have the log of two raised to the power four minus the log of five squared over the log of two raised to the power six minus the log of five raised to the power three. And now, since within the arguments of our logarithm we have exponents, we can use the power rule for logarithms. This tells us that the logarithm to the base 𝑎 of 𝑏 raised to the power 𝑐 is equal to 𝑐 times the logarithm to the base 𝑎 of 𝑏. That is, we bring the exponent to the front of our expression and multiply by it.
In our first term, for example, you bring the four down to the front and multiply by this, resulting in four multiplied by the logarithm of two. Similarly, for our second expression, we have an exponent of two. So the two comes in front of the logarithm of five. In the denominator, the exponent of six multiplies the logarithm of two. And in the second term, in the denominator, the exponent of three multiplies logarithm of five. We therefore have four times the logarithm of two minus two times the logarithm of five divided by six times the logarithm of two minus three times the logarithm of five.
Now we notice that in our numerator, we have a common factor of two. And in our denominator, we have a common factor of three. And taking these common factors outside some parentheses, we have two multiplied by two log two minus log five over three multiplied by two log two minus log five. And now we see in both numerator and denominator, we have a common factor of two log two minus log five. And dividing both numerator and denominator by this common factor, we have two multiplied by one over three multiplied by one, that is, two over three.
Hence, we find that the logarithm of 16 minus the logarithm of 25 divided by the logarithm of 64 minus the logarithm of 125 is equal to two over three.
