Let x log3 1243 State the corresponding exponential form of

Let x = log_3 1/243. State the corresponding exponential form of this logarithmic equation. Determine the numerical value of log_3 1/243 in simplest form.

Solution

x = log₃ (1/243)

==> (1/243) = 3^x      since if x = log_a b ==> b = a^x

==> 1/3⁵ = 3^x

==> 3^x 3⁵ = 1

==> 3^x+5 = 1           since a^m aⁿ = a^m+n

Hence x = log₃ (1/243) in exponential form ==> 3^x+5 = 1

b) log₃ (1/243)

==> log₃ 1 - log₃ 243 since log (a/b) = log a - log b

==> 0 - log₃ 3⁵    since log_b 1 = 0

=> - 5 log₃ 3          since log a^b = b log a

==> -5 (1)           since loa_a a = 1

==> -5

Hence log₃ (1/243) = -5