natural algorithms

Question

Express in terms of common or natural algorithms: log₅24

Answer 1

using ln(x) for natural logarithm for x

k=log₅24

5^k=24

ln5^k=ln24

k*ln5=ln24 ,,,,,,,logarithm rule for exponents

k=ln24/ln5

k=k

therefore

log₅24=ln24/ln5

Answer 2

log₅24 =ln24/ln5

for natural logs, base e

5^2 = 25, so the log of 24 base 5 should be slightly less than 2

= about 1.98

=log24/log5

for common logs (an implicit base 10. same as log₁₀24/log₁₀5)

general formula is log_ba = log_ca/log_cb

for any a, b, and c

change of base formula

in this given example b=5, a=24,

c is e for natural logs, c=10 for common logs