1. ## [SOLVED] Showing?

Given that $\displaystyle 2^a = 3^b = 18^c$, show that ab = c(b + 2a)

2. Originally Posted by mark1950
Given that $\displaystyle 2^a = 3^b = 18^c$, show that ab = c(b + 2a)
$\displaystyle 2^a=3^b$

$\displaystyle aln(2)=b ln(3)$

$\displaystyle ln(3)=\frac{a ln(2)}{b}$ .....(1)

$\displaystyle 3^b=18^c$

$\displaystyle 3^b=(2(3)^2)^c$

$\displaystyle b ln(3) = c ln(2 (3^2))$

$\displaystyle b ln(3) = c(ln(2) + 2ln(3) )$ since $\displaystyle ln(AB) = ln(A) + ln(B)$

sub the value of ln(3) from (1)

$\displaystyle b\left(\frac{a ln(2)}{b}\right) = c(ln(2) + 2\left(\frac{a ln(2)}{b}\right)$

$\displaystyle b\left(a ln(2)\right) = cbln(2) + 2c\left(a ln(2)\right)$ by multiply with b

$\displaystyle b\left(a \right) = cb + 2c\left(a \right)$ division by ln(2)

$\displaystyle ba = c(b+2a)$

3. What does ln means? Can I use log instead of ln? Will it be the same?

4. Originally Posted by mark1950
$\displaystyle \ln$ denotes the natural log, but for this any base of logarithm can be used. \log often denotes a logarithm to the base $\displaystyle 10$, but just as often denotes a natural logarithm or a logarithm to an unspecified base, so yes you can use $\displaystyle \log$.