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What is the value of log, base, 5, start root, start index, 4, end index, 5, end rootlog 5​ 4 5​ ?

Question

What is the value of log554log545 \log_{5} \sqrt[4]{5} \, \log_{5} 4 \, 5 ?

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Solution

Sure, let's solve the problem step by step.

We need to find the value of log554 \log_5 \sqrt[4]{5} .

Step 1: Express the fourth root in exponential form. 54=51/4 \sqrt[4]{5} = 5^{1/4}

Step 2: Substitute this into the logarithm. log5(51/4) \log_5 (5^{1/4})

Step 3: Use the logarithm power rule, which states logb(ac)=clogb(a) \log_b (a^c) = c \log_b (a) . log5(51/4)=14log5(5) \log_5 (5^{1/4}) = \frac{1}{4} \log_5 (5)

Step 4: Simplify using the fact that logb(b)=1 \log_b (b) = 1 . log5(5)=1 \log_5 (5) = 1

Step 5: Substitute back into the equation. 14log5(5)=14×1=14 \frac{1}{4} \log_5 (5) = \frac{1}{4} \times 1 = \frac{1}{4}

So, the value of log554 \log_5 \sqrt[4]{5} is 14 \frac{1}{4} .

This problem has been solved

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