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 $\displaystyle{ 100(\frac{1}{5})^{\displaystyle{ \frac{x}{4}}}=4.$

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

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

$\displaystyle{ \frac{x}{4}}=2$, 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 $\displaystyle{ \frac{2}{\ln 6}.$

So, the function would be $h(x)=\frac{2}{\ln 6} \ln (x+5).$


Answers:

 1. 1

 2. 8

 3. $h(x)= \frac{2}{\ln 6}\ln (x+5).$