Question

Kim solved the two logarithmic equations below.Analyze her work and make any corrections to her work to help her have the correct solutions.Explain any errors identified.

Answer 1

a. The given equation is:

`-5+10^x=25`

The first step is adding 5 to both sides:

`\begin{gathered} -5+10^x+5=25+5 \\ 10^x=30 \end{gathered}`

The next step is to apply logarithm to both sides (Kim didn't do it and it is a mistake):

`\log 10^x=\log 30`

Apply the properties of logarithms:

`x\log 10=\log 30`

Divide both sides by log 10 and solve for x:

`\begin{gathered} (x\log10)/(\log10)=(\log 30)/(\log 10) \\ \text{Simplify} \\ x=(\log30)/(\log10) \\ x=1.47712 \end{gathered}`

The mistake that Kim made was not using the logarithm in the second step.

b. Solve for x:

`\log _5x=2`

We can relate logarithms to exponential function as follows:

`\log _bn=a\Leftrightarrow b^a=n`

Then, we can say that:

`\log _5x=2\Leftrightarrow5^2=x`

Kim applied the wrong formula and made a^b=n, that was the mistake.

And now we can solve for x:

`\begin{gathered} x=5^2 \\ x=5\cdot5 \\ x=25 \end{gathered}`