Question

1. Find the exact solution to the equation. 6-log8(x+7) = 5

a) x = 15 b) x = 1 c) x = 2 d) x = -1

2. Find the exact solution to the equation. (5 points)
one hundred times one fifth to the power of quantity x divided by four equals four.


a) x = one divided by two
b) x = 8
c) x = 2
d) x = 9

3. Determine the function which corresponds to the given graph.
a natural logarithmic function crossing the x axis at negative four and going through the point 1,2.

The asymptote is x = -5

Answer 1

1. The equation is
`6-\log_8(x+7) = 5.`

Rearranging, we have
`\log_8(x+7) = 1.` We know that (x+7) must be 8, since
`\log_88=1`, so we set the equation:

x+7=8, which means that x=1.

(quick check:
`6-\log_8(1+7) = 6-\log_8(8)=6-1=5.`

2.
The equation is
`\displaystyle{ 100((1)/(5))^{\displaystyle{ (x)/(4)}}=4.`

Dividing both sides by 100, the right hand side becomes 1/25, which can be written as
`( (1)/(5) )^2`.

Now that both exponential expressions have the same base, we set the exponents equal:


`\displaystyle{ (x)/(4)}=2`, which gives us x=8.


3.

The most well known natural logarithmic function is
`f(x)=\ln x` (that is:
`f(x)=\log_ex`.)

The asymptote of this function is the line x=0. This means that we have to shift
`f(x)=\ln x` 5 units left, to get:


`g(x)=\ln (x+5).`

Now, at 1, this function takes the value
`g(1)=\ln (1+5)=\ln (6).` To make the function pass through (1, 2), we could multiply by
`\displaystyle{ (2)/(\ln 6).`

So, the function would be
`h(x)=(2)/(\ln 6) \ln (x+5).`


Answers:

1. 1

2. 8

3.
`h(x)= (2)/(\ln 6)\ln (x+5).`