ACT Math · Exponential and logarithmic functions

What is (4) + (25)?

What is log⁡(4)+log⁡(25)\log(4) + \log(25)?

Answer and explanation

Answer: J: 2

The product rule for logarithms says log⁡a+log⁡b=log⁡(ab)\log a+\log b=\log(ab). So log⁡4+log⁡25=log⁡(4⋅25)=log⁡100\log4+\log25=\log(4\cdot25)=\log100. A common logarithm has base 1010, and 102=10010^2=100, so log⁡100=2\log100=2, choice J. Choice H (2929) adds the numbers inside the logarithms, 4+254+25, which is not a valid rule. Choice G (100100) is the argument of the logarithm rather than its value. Choice F (11) would be log⁡10\log10. The product rule turns a sum of logarithms into the logarithm of a product, so look for numbers that multiply to a power of the base.

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