09.5 - Solving Logarithmic Equations

• 15

• 10

• 20

• 5

Answer explanation

To solve \( \log_8 5 + \frac{1}{2} \log_8 9 = \log_8 x \), use properties of logarithms. Combine the logs: \( \log_8 (5 \cdot 9^{1/2}) = \log_8 x \). This simplifies to \( \log_8 (5 \cdot 3) = \log_8 x \), giving \( x = 15 \).

• 2, 6

• 6

• 2, 4

• 2

Answer explanation

To solve the equation, use properties of logarithms to combine terms. After simplification, you find the quadratic equation (2x-6)(x-2)=0, giving solutions x=2 and x=6. Thus, the correct answers are 2 and 6.

• 3

• 2

• 5

• 1

Answer explanation

To solve \( \ln10 - \frac{1}{2}\ln25 = \ln x \), simplify to \( \ln10 - \ln5 = \ln x \). This gives \( \ln\frac{10}{5} = \ln x \), so \( x = 2 \). Thus, the correct answer is 2.

• 10

• 50

• 100

• 25

Answer explanation

To solve \(\frac{1}{2}\ln x=3\ln5-\ln x\), first combine the logarithms: \(\frac{1}{2}\ln x + \ln x = 3\ln5\). This simplifies to \(\frac{3}{2}\ln x = 3\ln5\). Solving gives \(x = 25\). Thus, the correct answer is 25.

• 3

• 9

• 12

• 6

Answer explanation

To solve \ln x - \frac{1}{2}\ln 3 = \frac{1}{2}\ln(x+6), combine the logarithms and exponentiate. This leads to the equation x = 6, which satisfies the original equation. Thus, the correct answer is 6.

• x = {3, 6}

• x = 3

• x = {6, 9}

• x = 1

Answer explanation

To solve \ln x - \frac{1}{2}\ln 9 = \frac{1}{2}\ln(x-2), combine logarithms and exponentiate. This leads to the quadratic equation x^2 - 9x + 18 = 0, which factors to (x-3)(x-6)=0. Thus, x = {3, 6}.

• 10

• 3

• 5

• 1

Answer explanation

To solve, simplify the equation: 3ln(e/√5) = 3 - ln(x). This leads to ln(x) = 3 - 3ln(e/√5). Solving gives x = 5, which is the correct answer.

Tags

CCSS.HSF.BF.B.5

Answer explanation

To solve, first simplify: 5ln(e) - 5ln(2^(1/5)) = 1 - ln(x). This gives 5 - ln(2)/5 = 1 - ln(x). Rearranging leads to ln(x) = 1 - 5 + ln(2)/5 = ln(2)/5 - 4. Thus, x = e^(ln(2)/5 - 4) = 2/e^4.

Tags

CCSS.HSF.BF.B.5

• x = 5

• x = 25

• x = 10

• x = 2

Answer explanation

To solve, rewrite the equation: \(2\log_4x = \log_425\) becomes \(\log_4x^2 = \log_425\). Thus, \(x^2 = 25\) leads to \(x = 5\) (since x must be positive). Therefore, the correct answer is x = 5.

• x = 16

• x = 8

• x = 4

• x = 32

Answer explanation

To solve \(\log_2 x = 4\), rewrite it in exponential form: \(x = 2^4\). Calculating this gives \(x = 16\). Thus, the correct answer is \(x = 16\).

Tags

CCSS.HSF.BF.B.5