Solve equation for x

Question

Question: $$3 \cdot 2^{x/4} = 10 \cdot 3^{x/6}$$

My answer:

So I thought I could take the natural logarithm on both sides, making the equation:

$$3x/4 \cdot \ln2= 10x/6 \cdot \ln3$$

But then x cancels on both sides so I know thats the wrong way

Answer 1

Given that you have $$3 \cdot 2^{x/4} = 10 \cdot 3^{x/6},$$ you may take the natural logarithm of both sides like so: $$\ln\left(3 \cdot 2^{x/4}\right) = \ln\left(10 \cdot 3^{x/6}\right).$$ However, recall that $\ln(uv) = \ln(u) + \ln(v)$. Hence $$\ln\left(3 \right) + \ln\left( 2^{x/4}\right) = \ln\left(10 \right) + \ln\left( 3^{x/6}\right).$$ This is where your mistake occurs. From here, you proceed via the power rule for logarithms.