Question

I need help with this question please1) Picture2) In(2x) + ln(7) = 4

Answer 1

1) We must solve for x the following equation:

`e^(5x)=25.`

To solve this equation, we take the natural logarithm to both sides of the equation:

`\begin{gathered} \ln (e^(5x))=\ln (25), \\ 5x\cdot\ln e=\ln 25. \end{gathered}`

Now, we use the following results:

`\begin{gathered} \ln e=1, \\ \ln (25)=\ln (5^2)=2\cdot\ln 5. \end{gathered}`

Replacing these results in the equation above, we have:

`5x=2\cdot\ln 5.`

Solving for x, we get:

`x=(2)/(5)\cdot\ln 5\cong0.64.`

2) We must solve for x the following equation:

`\ln (2x)+\ln (7)=4.`

To solve this problem, we isolate the part that involves the x:

`\ln (2x)=4-\ln (7)\text{.}`

Now, using the following property:

`\ln y=z\rightarrow y=e^z\text{.}`

with:

`\begin{gathered} y=2x, \\ z=4-\ln 7. \end{gathered}`

we have:

`\ln (2x)=4-\ln 7\rightarrow2x=e^(4-\ln 7).`

Solving the last equation for x, we get:

`x=(1)/(2)\cdot e^(4-\ln 7)\cong3.90.`

Answers

1) The value of x that solves the first equation is 0.64 to two decimal places.

2) The value of x that solves the second equation is 3.90 to two decimal places.

Review of the base of a logarithm

We can define the logarithm in base a through the following equations:

`\begin{gathered} \log _aa=1, \\ \log _aa^x=x\cdot\log _aa=x\cdot1=x\text{.} \end{gathered}`

When we use as a base the Euler number e ≅ 2.718, the logarithm is called "natural" and we use the following notation for it:

`_{}\log _e=\ln .`

With this notation, we have the following properties:

`\begin{gathered} \ln e=1, \\ \ln e^x=x\cdot\ln e=x\cdot1=x\text{.} \end{gathered}`