Question

How you you cancel logarithms on either side of an equation

Answer 1

`\log_7(3x^3-18)=3\log_7(2)`

In your first step, you incorrectly applied the power property, which says


`\log_b(a^c)=c\log_b(a)`

But
`a` is a single term, not a sum/difference of terms.

You can however use the reverse of this rule to rewrite the right hand side:


`\log_7(3x^3-18)=\log_7(2^3)=\log_7(8)`

Now, on both sides you're taking the same logarithm of two expressions, and the logarithms are equal, which means the two expressions must also be equal. But more explicitly, we can "cancel" the logarithms by taking both sides of the equation to be powers of 7:


`7^(\log_7(3x^3-18))=7^(\log_7(8))`

and this invokes the property that


`b^(\log_b(a))=a`

so that the equation reduces to


`3x^3-18=8\implies3x^3=26\implies x^3=\frac{26}3\implies x=\sqrt[3]{\frac{26}3}`