Question 1

Which of the following is the correct solution to the equation $$\log_6$ x = 2$$?

Accepted Answer

$$x = 36$$

Question 2

If $$\log_{20}(x+9) + \log_{20}(x+1) = 1$$, what is the solution set?

Accepted Answer

$$\{1\}$$

Question 3

Solve the exponential equation $$3^x = 7. $$What is the exact solution?

Accepted Answer

$$x = \log_3$ 7$$

Question 4

Which property allows you to combine $$\log_a M + \log_a N$$ into a single logarithm?

Accepted Answer

$$\log_a (MN)$$

Question 5

What is the solution to $$\ln x + \ln(x-1) = \ln(x+15)$$, given the domain restrictions?

Accepted Answer

$$x = 5$$

Question 6

Solve $$5 \cdot 4^x = 3$$ for $$x. $$What is the exact solution?

Accepted Answer

$$x = \frac{\ln 3 - \ln 5}{\ln 4}$$

Question 7

Which of the following is a valid step when solving $$\ln x + \ln(x+8) = 3$$?

Accepted Answer

$$\ln[x(x+8)] = 3$$

Question 8

A student solves $$\log_3$ x^2$ = 4$$ and finds $$x = 9$$ and $$x = -9. $$Which caution should they consider?

Accepted Answer

Check if the negative solution causes the argument to be negative or zero.

Question 9

Solve the exponential equation $$7^{x-3}$$ = $$2^{2x+1}. $$What is the exact solution for $$x$$?

Accepted Answer

$$x = \frac{3 \ln 7 + \ln 2}{\ln 7 - 2 \ln 2}$$

Question 10

Which method is appropriate for solving $$x + e^x = 3$$ when an exact algebraic solution is not possible?

Accepted Answer

Graphing the functions and finding the intersection

Question 11

If $$\log_a M = \log_a N$$, what must be true about $$M$$ and $$N$$?

Accepted Answer

$$M = N$$

Question 12

What is the solution to $$9^x + 3^x - 12 = 0$$?

Accepted Answer

$$x = 1$$

Question 13

Which equation is equivalent to $$\log_a x^r = r \log_a x$$ only when $$x \u003e 0$$?

Accepted Answer

$$\log_a x^r = r \log_a x$$ only when $$x \u003e 0$$

Question 14

What is the solution to $$\ln x + \ln(x+6) = 4$$ using the quadratic formula?

Accepted Answer

$$x = -3 + \sqrt{9 + e^4$}$$

Question 15

Which of the following is the correct change of base formula for $$\log_a b$$?

Accepted Answer

$$\log_a b = \frac{\ln b}{\ln a}$$

Question 16

If $$a^u = a^v$$ and $$a \u003e 0$$, $$a 
eq 1$$, what can you conclude?

Accepted Answer

$$u = v$$

Question 17

A medical researcher models the growth of bacteria with $$N(t) = N_0$ e$$^{kt}$$. If the population triples in 5 hours, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 3}{5}$$

Question 18

Which of the following is NOT a valid domain restriction for $$\log_a(x-2)$$?

Accepted Answer

$$x 
eq 2$$

Question 19

If $$\ln x + \ln(x+8) = 3$$, what is the approximate solution rounded to three decimal places?

Accepted Answer

$$x \approx 2.007$$

Solving Logarithmic Equations: log₆x = 2

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