How to add and subtract logarithms with different bases?

Question

How can I add and subtract logarithms with different bases? For example, how do I solve the expression log₂(32) - log₄(16) + log₃(27)? I would appreciate a detailed explanation as well. Thank you!

Aisha Denesik · Accepted Answer

There are 2 ways to do this particular problem. The main purpose of this exercise is to teach you the following rule of logarithms:

log[a](a^b) = b

Now, let’s look at your problem:

log[2](32) – log[4](16) + log[3](27)

Let’s rewrite this a bit:

32 = 2^5

16 = 4^2

27 = 3^3

log[2](2^5) – log[4](4^2) + log[3](3^3) =>

5 * log[2](2) – 2 * log[4](4) + 3 * log[3](3) =>

5 * 1 – 2 * 1 + 3 * 1 =>

5 – 2 + 3 =>

6

Now, what’s the second way? Well, that involves something known as the “Change of Base” formula

log[a](b) =>

log[c](b) / log[c](a)

For instance, log[2](32) can be rewritten as log[10](32) / log[10](2)

log[2](32) – log[4](16) + log[3](27) =>

log(32) / log(2) – log(16) / log(4) + log(27) / log(3) =>

log(2^5) / log(2) – log(4^2) / log(4) + log(3^3) / log(3) =>

5 * log(2) / log(2) – 2 * log(4) / log(4) + 3 * log(3) / log(3) =>

5 – 2 + 3 =>

6

We could also try and get a common denominator and solve that way, but I have a sick feeling in my gut that tells me that that particular method would be monstrous to behold.

Anonymous · Answer

Subtract Logarithms

Johanna Ortiz · Answer

log2 32 – log4 16 + log3 27
 = log2 2^5 – log4 4^2 + log3 3^3
 = 5 – 4 + 3
 = 4
The explanation is log_b(b^x) = x log_b(b) = x*1 = x

Anonymous

How to add and subtract logarithms with different bases?

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