Question

math 102 exam 3 version oc summer 2006

1. (8 points) simplify fully.

(a) (e^{ln 3x})
(b) (log 10^{7x})
(c) (log_2 8^{3x})
(d) (6^{log_6 2x})

1. (6 points) evaluate.

(a) (log 10000)
(b) (log_{0.5} 16)
(c) (log_6 36)

1. (5 points) convert to an exponential equation.

(log_4 64 = 3)

Explanation:

Step1: Simplify 1(a) using inverse properties

$$e^{\ln 3x} = 3x$$

Step2: Simplify 1(b) using common logarithm properties

$$\log 10^{7x} = 7x$$

Step3: Simplify 1(c) by rewriting the base

$$\log_2 8^{x} = \log_2 (2^3)^x = \log_2 2^{3x} = 3x$$

Step4: Simplify 1(d) using inverse properties

$$6^{\log_6 2x} = 2x$$

Step5: Evaluate 2(a) using powers of 10

$$\log 10000 = \log 10^4 = 4$$

Step6: Evaluate 2(b) using powers of 2

$$\log_{0.5} 16 = \log_{2^{-1}} 2^4 = -4$$

Step7: Evaluate 2(c) using fractional exponents

$$\log_{216} 36 = \log_{6^3} 6^2 = \frac{2}{3}$$

Step8: Convert 3 to exponential form

$$\log_4 64 = 3 \implies 4^3 = 64$$

Answer:

1. (a) \(3x\)

(b) \(7x\)
(c) \(3x\)
(d) \(2x\)

1. (a) \(4\)

(b) \(-4\)
(c) \(\frac{2}{3}\)

1. \(4^3 = 64\)