Question

Solve for x by using log. Rounded to hundredth.


`10^(4x-5)=e^(3x)`

Answer 1

`x=1.85`

Explanation:

The given exponential equation is;

`10^(4x-5)=e^(3x)`

We take logarithm of both sides to base
`e`.

`\ln 10^(4x-5)=\ln e^(3x)`

`(4x-5)\ln 10=3x\ln e`

`(4x-5)\ln 10=3x`

Expand the left hand side;

`4x\ln 10-5\ln 10=3x`

Group like terms

`4x\ln 10-3x=5\ln 10`

`(4\ln 10-3)x=5\ln 10`

`x=(5\ln 10)/(4\ln 10-3)`

`x=1.85`

Answer 2

Answer:
`x=1.85`

Explanation:

Given the expression
`10^(4x-5)=e^(3x)`, apply natural logarithm to both sides. Then:

`ln(10)^(4x-5)=ln(e)^(3x)`

Remember that according the the properties of logarithms:

`ln(a)^m=mln(a)`

`ln(e)=1`

Then:

`(4x-5)ln(10)={3x}`

Apply distributive property and solve for "x". Then you get:

`4x*ln(10)-5ln(10)=3x\\\\4x*ln(10)-3x=5ln(10)\\\\x(4ln(10)-3)=5ln(10)\\\\x=(5ln(10))/(4ln(10)-3)\\\\x=1.85`