8) solve the logarithmic equation. be sure to check for extraneous solutions.\n5·log₂2 = log₂8 + log₂(x…

Problem 8

Explanation:

Step1: Simplify left - hand side

Use the property (\log_a a = 1). So, (5\cdot\log_2 2=5\times1 = 5).

Step2: Simplify right - hand side

Use the property (\log_a M+\log_a N=\log_a(MN)). Then (\log_2 8+\log_2(x - 2)=\log_2(8(x - 2))). Since (8 = 2^3), (\log_2 8=3), and the equation becomes (5=\log_2(8(x - 2))).

Step3: Convert to exponential form

Using the property (y=\log_a x\Leftrightarrow x = a^y), we have (2^5=8(x - 2)). Since (2^5=32) and (8(x - 2)=8x-16), the equation is (32 = 8x-16).

Step4: Solve for (x)

Add (16) to both sides: (32 + 16=8x), so (48 = 8x). Then (x = 6).

Step5: Check for extraneous solutions

For the original equation (\log_2(x - 2)), when (x = 6), (x-2=4>0).

Answer:

(x = 6)

Problem 9

Explanation:

Step1: Take the natural logarithm of both sides

Given (7^{x - 4}=26), take (\ln) on both sides: (\ln(7^{x - 4})=\ln(26)).

Step2: Use the power rule of logarithms

Using the property (\ln(a^b)=b\ln(a)), we get ((x - 4)\ln(7)=\ln(26)).

Step3: Solve for (x)

First, divide both sides by (\ln(7)): (x - 4=\frac{\ln(26)}{\ln(7)}). Then (x=\frac{\ln(26)}{\ln(7)}+4). Using a calculator, (\ln(26)\approx3.258), (\ln(7)\approx1.946), (\frac{\ln(26)}{\ln(7)}\approx1.674), and (x\approx1.674 + 4=5.674).

Answer:

(x=\frac{\ln(26)}{\ln(7)}+4\approx5.67)