Question

How do you solve `4^x * 5^(4x+3) = 10^(2x+3)`?

Answer 1

`x~~0.65`

Explanation:

`1`. Start by taking the logarithm of both sides, since the bases are not the same on the left and right sides of the equation.

`4^x*5^(4x+3)=10^(2x+3)`

`log(4^x*5^(4x+3))=log(10^(2x+3))`

`2`. Use the log property, `log_color(purple)b(color(red)m*color(blue)n)=log_color(purple)b(color(red)m)+log_color(purple)b(color(blue)n)` to simplify the left side of the equation.

`log(4^x)+log(5^(4x+3))=log(10^(2x+3))`

`3`. Use the log property, `log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)`, to rewrite both sides of the equation.

`xlog(4)+(4x+3)log(5)=(2x+3)log(10)`

`4`. Expand the brackets.

`xlog(4)+4xlog(5)+3log(5)=2xlog(10)+3log(10)`

`5`. Bring all terms with the variable, `x`, to the left side of the equation and all terms without to the right side.

`xlog(4)+4xlog(5)-2xlog(10)=3log(10)-3log(5)`

`6`. Factor out `x` from the terms on the left side.

`x(log(4)+4log(5)-2log(10))=3log(10)-3log(5)`

`7`. Solve for `x`.

`x=(3log(10)-3log(5))/(log(4)+4log(5)-2log(10))`

`color(green)(|bar(ul(color(white)(a/a)x~~0.65color(white)(a/a)|)))`

Answer 2

`0.646`

Explanation:

`4^x*5^(4x+3)=10^(2x+3)`

`=>4^x*5^(4x)*5^3=10^(2x)*10^3`

`=>(4*5^4)^x*5^3=(10^2)^x*10^3`

Dividing both sides by `5^3*(10^2)^x`

`=>((4*5^4)/10^2)^x=10^3/5^3`

`=>((cancel4*cancel5xxcancel5xx5^2)/(cancel10xxcancel10))^x=1000/125=8=2^3`

Taking`log_10` on both sides and using property `logm^n=nlogm` we get
`=>log_10 5^(2x)=log_10 2^3`
`=>2x*log_10 5=3*log_10 2`

Dividing both sides by `(2*log_10 5)`

`=>x=(3*log_10 2)/(2*log_10 5)=0.646`