Video Transcript
What is the value of the logarithm to the base three of 32 over the logarithm to the base three of 256?
To find the value of this expression, we’re going to apply our knowledge of powers of two together with the power rule for logarithms. We first note that the argument of the logarithm to the base three of 32, that is, 32, is equal to two raised to the power five. Similarly, the argument of the logarithm in the denominator, that is, 256, is equal to two raised to the power eight. And rewriting our expression, this gives us log to the base three of two raised to the power five over log to the base three of two raised to the power eight.
And now recalling the power rule for logarithms, which says that the logarithm to the base 𝑎 of 𝑏 raised to the power 𝑐 is equal to 𝑐 multiplied by the logarithm to the base 𝑎 of 𝑏, that is, we bring the exponent to the front of the expression and multiply by it, in our case, our exponents are five and eight. Hence, bringing our exponents down, we have five multiplied by the logarithm to the base three of two divided by eight multiplied by the logarithm to the base three of two. So now we have five times the logarithm to the base three of two divided by eight times the logarithm to the base three of two.
Now notice that in our numerator and denominator we have a common factor. That’s the logarithm to the base three of two. And dividing both numerator and denominator by this, we have five multiplied by one divided by eight multiplied by one, that is, five divided by eight. Hence, the logarithm to the base three of 32 divided by the logarithm to the base three of 256 is equal to five over eight.
