Solve for x 5^2x = 3(7^3x-2)

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quirozd | High School Teacher | (Level 3) Adjunct Educator

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I assume that you mean:

`5^(2x)=3(7^(3x-2))`

First take the log of both sides:

`log(5^(2x))=log(3(7^(3x-2)))`

Then use the properties of logarithms to help us:

`log_b(x^n)=nlog_b(x)`

and

`log_b(xy)=log_b(x)+log_b(y)`

and

`log_b(x/y)=log_b(x)-log_b(y)`

So our equation becomes:

`2xlog(5)=log(3)+(3x-2)(log7)`

Now, we trudge through the math to isolate x on one side:

2log(5)x=log(3)+3log(7)x-2log(7)

2log(5)x-3log(7)x=log(3)-2log(7)

x(2log(5)-3log(7))=log(3)-2log(7)

:. x = (log(3)-2log(7))/(2log(5)-3log(7))

Either punch the above into a calculator, or further reduce using the log properties in reverse.

`x = log(3/7^2)/log(5^2/7^3)=log(3/49)/log(25/343)`

`x ~~ 1.07`

Check our answer:

`5^(2*(1.07))=3(7^(3(1.07)-2))`

`30.97487349=30.97487349`

(we use the original result in our calculator to check, rather than the rounded one)

Answer: `x~~1.07`

Sources:

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