Hello Candice, there are several ways that c can be expressed in terms of a and b. If you could provide the answer, then we can provide the solution to match the answer. However, I do not really encourage that because you will end up working to match the provided answer rather than understanding the concept.

Thanks Mr Wong, the answer reflected in the answer key is ab/a+b

Got it! Thanks. Check the solution. Let me know if you have further questions.

The phrasing of the question already suggests/infers that c is to be expressed in terms of a and b only, i.e convert the terms to a and b as completely as possible. So the student should aim for that regardless whether the answer is given or not

101^a = 103^b = 10403^c

Take ln on all sides,

ln 101^a = ln103^b = ln10403^c

a ln101 = b ln103 = c ln10403


c ln10403 = b ln103

c (ln(101 x 103)) = b ln103

c (ln101 + ln103) = b ln103

c = b ln103 / (ln101 + ln103)

c = b / (ln101/103 + 1)

c = b / (b/a + 1)

(a ln101 = b ln103 → ln101/ln103 = b/a)

c = ab/(a+b) (multiply both numerator and denominator by a)

Thank you Mr J. Your advised working solution looks good :)

Have a good day ahead and take care.

Welcome

Alternatively,

101^a = 103^b = 10403^c

101^a = 103^b = (101 x 103)^c

101^a = 103^b = 101^c 103^c

Now 101^a = 103^b

So 101^c = 101^(ac/a) = 103^(bc/a) ①


And 101^c 103^c = 103^b

Sub in ①,

103^(bc/a) 103^c = 103^b

103^(bc/a + c) = 103^b

So bc/a + c = b

bc + ca = ab

c(a + b) = ab

c = ab/(a + b)