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How to Add Logarithms with different bases?

Log4 32 + Log2 32
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4 Answers

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Francesco Paucek

Feb 20, 2025

In general, you need convert one of the logarithms to the same base as the other one by using the relationship: log_a(b) * log_b(x) = log_a(x) or, written another way:log_b(x) = log_a(x)/log_a(b)where log_a and log_b mean the logarithms in base a and b, respectively.For instance:log_4(32) = log_2(32)/log_(4) but log_2(4) is 2 (i.e., 2^2 = 4), so:log_4(32) = (1/2)*log_2(32)You can now use this to carry out the addition:log_4(32) + log_2(32) = (1/2)log_2(32) + log_2(32)= 1.5*log_2(32) = 1.5*log_2(2^5) = 1.5*5 = 7.5In this case, however, there is an easier way. Note that 32 = 2^5, and 4 = 2^2, so 32 = 4^2.5. The log_2(2^5) = 5, and log_4(4^2.5) = 2.5, solog_4(32) + log_2(32) = 5 + 2.5 = 7.5... Show More
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Francesco Paucek

Feb 20, 2025

Use this formulation to calculate logarithm to any base log_b x=(log_k x )/(log_k b) So log_3 80 one=(log_10 80 one )/(log_10 3 )= 4 (you are able to calculate log to base 10 via using your calculator now) and in addition log_4 sixty 4=(log_10 sixty 4 )/(log_10 4 )=3 4+3=7(answer)... Show More
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