Question

Which two of the following equations use the inverse property of logarithms correctly? Select all that apply:

a) (5log₄(5))
b) (2ˡog₂(8))
c) (log₃(9²))
d) (ln(e²))

Answer 1

Equations b) 2log2(8) and d) ln(e2) both use the inverse property of logarithms correctly. In both, the base of the logarithm matches the base of the exponent, which means they 'cancel out', leading to a simplified answer.

Step-by-step explanation:

The student has asked about the inverse property of logarithms, which pertains to the concept that logarithmic and exponential functions are inverses of each other. Specifically, logb(bx) = x, where b is the base of the logarithm and the exponential function, and x is the exponent. Similarly, blogb(x) = x. Therefore, to identify equations that use the inverse property of logarithms correctly, we look for expressions where the base of the logarithm and the base of the exponent match, effectively canceling each other out.

Analysis of the Given Equations

• b) 2log2(8): This applies the inverse property correctly, as the base of the logarithm (2) matches the base of the exponent. Simplified, it would yield the exponent to which 2 must be raised to get 8, which is 3.
• d) ln(e2): This also correctly applies the inverse property. Here, e is the base of the natural logarithm and the exponential function. Simplified, it would yield 2, since eln(x) = x, and thus ln(e2) = 2.

Equations a) and c) do not demonstrate the inverse property correctly because the bases are not consistent within the expressions, thus they do not cancel out in the same way.