In order to solve a logarithm that appears in an exponent, you need to know all the logarithm rules including the sum of logarithms, product of logarithms, change of base rule, etc.
In order to solve a logarithm that appears in an exponent, you need to know all the logarithm rules including the sum of logarithms, product of logarithms, change of base rule, etc.
Solution steps:
\( \log_35x\times\log_{\frac{1}{7}}9\ge\log_{\frac{1}{7}}4 \)
To simplify the expression , we apply the power property of logarithms, which states:
Step 1: Identify the given expression: .
Step 2: Apply the power property of logarithms:
Step 3: Calculate :
Step 4: Substitute back into the logarithmic expression:
Therefore, the simplified expression is .
Comparing with the answer choices, the correct choice is:
Answer:
To solve this problem, let's simplify using logarithm rules.
This is a straightforward application of the power property of logarithms. By applying this property correctly, we've simplified the original expression correctly.
Therefore, the simplified form of is .
Answer:
To solve the problem , we need to express the number 8 as a power of a base that simplifies the logarithm. We can write 8 as , because 8 equals 2 multiplied by itself three times.
Let's use the power property of logarithms, which is:
Applying this property to , we have:
Using the power property, this becomes:
Therefore, the expression for in terms of is:
.
Answer:
To solve this problem, we'll follow the steps outlined:
Therefore, the rewritten expression for using logarithm rules is .
This matches choice 4 from the provided options.
Answer:
7\log_42<\log_4x
To solve the inequality , we will follow these steps:
Let's now proceed with these steps:
Step 1: Using the power property of logarithms, we have . This step simplifies the multiplication into a single logarithmic term.
Step 2: Using substitution in the inequality, we write it as .
Step 3: Since logarithms are one-to-one functions, we can conclude that if , then . This results from the property where the bases are equal.
Therefore, the solution to the inequality is .
Answer:
2^7 < x