Hi, my name is Chris and i have a question concerning natural logarithms. I am currently studying year 12 math in Australia. I have a series of questions that are an extension from what we are covering in the text book. I was hoping you could explain how do so one of the questions so i can do the rest myself. Here is an example of one of the questions:

Given that log_e y = 0.3log_ex+1.2 show that y=3.32 x^0.3

Yours Truly

Chris

Hi Chris,

To solve for y you need to use two of the properties of logarithms. They are

log(a b) = log(a) + log(b) and
log(b^a) = a log(b)

The second of these properties I can use immediately to write

0.3 log_ex = log_e(x^0.3 )

and hence

log_e y = 0.3log_ex+1.2 = log_e(x^0.3 ) + 1.2

I could now use the first property of logarithms I mentioned above if the number 1.2 were a logarithm. Is there a number a so that 1.2 = log_e(a)? If so you can write

log_e y = log_e(x^0.3 ) + 1.2 = log_e(x^0.3 ) + log_e(a) = log_e(a x^0.3)

and hence

y = a x^0.3

What is a?
Penny