change-of-base formula

Question

Use the change-of-base formula to convert x=log₃90 to common logarithms. Then use a calculator to round the value of x to the nearest thousandth.

Answer 1

Hi Yittel,

The change-of-base formula is another way of solving log problems.

This is the formula using just letters, no numbers:

log_ab = (log_cb)/(log_ca)

a = the original base of the log

b = the number you are taking the log to.

c = this new base can be any number, it would not affect the result because both c are in the numerator and denominator. If you were to use a calculator and write that fraction, the calculator would assume that c = 10, because just typing log 16, for example, is the same as log₁₀16.

So, let us attempt the problem to find the value of x, to the nearest thousandths place.

x = log₃90 = (log 90)/(log 3) <-- Input this fraction into your calculator.

x = 4.095903274 ~ 4.096 <-- After rounding to the nearest thousandths place

Hope this helps, and if you would like more help, reply back to this response!

Answer 2

log90 to the base 3 = log90/log3 with base = 10

x =about 1.95424/.477121= 4.0959 = 4.096