Simplify: -3(ln4/ln5 - log₅7 + 1/log₆5) Logarithmic Expression

Question

3(ln4ln5log57+1log65)= -3(\frac{\ln4}{\ln5}-\log_57+\frac{1}{\log_65})=

Video Solution

Solution Steps

00:14 Let's begin by solving this problem!
00:18 First, we'll use the formula for dividing logarithms.
00:23 Take the log of the top number, divided by the bottom number.
00:28 Now, let's apply this division formula in our exercise.
00:43 Next, for one over a log, take the inverse of that log.
00:53 For subtracting logs, we find the log of their division.
01:03 Let's put these formulas into action in our exercise!
01:23 Add logs together to get the log of their multiplication.
01:33 Let's apply this addition formula now.
02:03 For logs with exponents, raise the number to that power.
02:08 Let's use this power formula in our exercise.
02:26 And that's how we solve the problem! Great job!

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Apply the change-of-base formula to ln4ln5\frac{\ln 4}{\ln 5}.

  • Step 2: Apply the reciprocal property to 1log65\frac{1}{\log_6 5}.

  • Step 3: Use the subtraction property of logs to simplify the expression.

  • Step 4: Combine the simplified logarithms and multiply by -3.

Now, let's work through each step:

Step 1: Using the change-of-base formula, we have ln4ln5=log54\frac{\ln 4}{\ln 5} = \log_5 4.

Step 2: Apply the reciprocal property to the third term: 1log65=log56\frac{1}{\log_6 5} = \log_5 6.

Step 3: Substitute into the expression: 3(log54log57+log56)-3(\log_5 4 - \log_5 7 + \log_5 6).

Step 4: Combine terms using the properties of logs: log54log57+log56=log5(4×67)\log_5 4 - \log_5 7 + \log_5 6 = \log_5 \left(\frac{4 \times 6}{7}\right).

Step 5: Simplify to get: log5(247)\log_5 \left(\frac{24}{7}\right).

Multiply by -3: 3(log5(247))=3log5(724) -3(\log_5 (\frac{24}{7})) = 3\log_5 \left(\frac{7}{24}\right) .

Therefore, the solution to the problem is 3log5724 3\log_5 \frac{7}{24} .

Answer

3log5724 3\log_5\frac{7}{24}


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