Question

Hello, help me please)

Answer 1

Given equation:

`10^(5x-2)=2^(8x-3)`

Take natural logs of both sides:

`\implies \ln 10^(5x-2)= \ln 2^(8x-3)`

`\textsf{Apply log Power law}: \quad \ln_ax^n=n\ln_ax`

`\implies (5x-2)\ln 10=(8x-3) \ln 2`

Expand brackets:

`\implies 5x\ln 10 - 2\ln 10=8x \ln 2 -3 \ln 2`

Collect like terms:

`\implies 5x\ln 10 - 8x \ln 2 =2\ln 10-3 \ln 2`

Factor left sides:

`\implies x(5\ln 10 - 8 \ln 2) =2\ln 10-3 \ln 2`

`\textsf{Apply log Power law}: \quad \ln_ax^n=n\ln_ax`

`\implies x(\ln 10^5 - \ln 2^8) =\ln 10^2- \ln 2^3`

`\textsf{Apply log Quotient law}: \quad \ln_a(x)/(y)=\ln_ax - \ln_ay`

`\implies x\left(\ln\left((10^5)/(2^8)\right)\right) =\ln\left((10^2)/(2^3)\right)`

Simplify:

`\implies x\left(\ln\left((3125)/(8)\right)\right) =\ln\left((25)/(2)\right)`

`\implies x=(\ln\left((25)/(2)\right))/(\ln\left((3125)/(8)\right))`

`\implies x=0.4232297737...`