Question

Solve for x using logarithm:
25^x-3(5^x)=0​

Answer 1

`x = 0.68`

Explanation:

We would like to find out the value of x using logarithms of the given equation .The equation is ,

`\longrightarrow 25^x - 3(5^x)=0\\`

Add
`3(5^x)` on both sides,

`\longrightarrow 25^x = 3(5^x)`

Using log to the base 10 on both sides, we have;

`\longrightarrow log_(10)(25^x) = log_(10)\{3(5^x)\}`

Recall that
`log(ab ) = log\ a + log\ b` .

`\longrightarrow log_(10)(25^x)=log_(10)3 + log_(10)5^x`

Recall the properties of logarithm as
`log\ a^b = b\ log\ a` .

`\longrightarrow xlog25 = log_(10)3 + xlog_(10)5`

Again we can rewrite it as ,

`\longrightarrow xlog(5^2)=log_(10)3+xlog_(10)5\\`

`\longrightarrow 2x\ log_(10)5 = log_(10)3+xlog_(10)5 \\`

`\longrightarrow 2x\ log_(10)5-x\ log_(10)5 = log_(10)5`

Simplify,

`\longrightarrow x\ log_(10)5=log_(10)3`

Divide both sides by log5 ,

`\longrightarrow x =(log_(10)3)/(log_(10)5)`

Put on the values of log 3 and log5 ,

`\longrightarrow x =(0.47)/(0.69)`

Simplify,

`\longrightarrow \underline{{\underline{\boldsymbol{ x = 0.68}}}}`

And we are done!