Power Property of Logorithms - Examples, Exercises and Solutions

Understanding Power Property of Logorithms

Complete explanation with examples

Power in logarithm

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:

  1. Take the logarithm with the same base on both sides of the equation.
    The base will be the original base - the one which the log power is applied to.
  2. Use the rule
    loga(ax)=xlog_a (a^x)=x
  3. Create a common base between the 22 equation factors in order to determine the solution.
  4. Solve the logs that can be solved and convert them to numbers.
  5. Insert an auxiliary variable into the problem TT if needed
  6. Go back to determine XX.
Detailed explanation

Practice Power Property of Logorithms

Test your knowledge with 6 quizzes

\( \log_35x\times\log_{\frac{1}{7}}9\ge\log_{\frac{1}{7}}4 \)

Examples with solutions for Power Property of Logorithms

Step-by-step solutions included
Exercise #1

3log76= 3\log_76=

Step-by-Step Solution

To simplify the expression 3log76 3\log_76 , we apply the power property of logarithms, which states:

alogbc=logb(ca) a\log_b c = \log_b(c^a)

Step 1: Identify the given expression: 3log76 3\log_76 .

Step 2: Apply the power property of logarithms:

3log76=log7(63) 3\log_76 = \log_7(6^3)

Step 3: Calculate 63 6^3 :

63=6×6×6=36×6=216 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216

Step 4: Substitute back into the logarithmic expression:

log7(63)=log7216 \log_7(6^3) = \log_7216

Therefore, the simplified expression is log7216\log_7216.

Comparing with the answer choices, the correct choice is:

log7216 \log_7216

Answer:

log7216 \log_7216

Video Solution
Exercise #2

2log38= 2\log_38=

Step-by-Step Solution

To solve this problem, let's simplify 2log382\log_3 8 using logarithm rules.

  • Step 1: Recognize the expression form
    The expression is of the form alogbca \cdot \log_b c, where a=2a = 2, b=3b = 3, and c=8c = 8.
  • Step 2: Apply the power property
    According to the power property of logarithms, 2log382 \cdot \log_3 8 can be simplified to log3(82)\log_3 (8^2).
  • Perform the calculation
    Calculate 828^2, which is 6464.
  • Step 3: Simplify further
    Therefore, we have log364\log_3 64.

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 2log382\log_3 8 is log364\log_3 64.

Answer:

log364 \log_364

Video Solution
Exercise #3

log68= \log_68=

Step-by-Step Solution

To solve the problem log68 \log_6 8 , we need to express the number 8 as a power of a base that simplifies the logarithm. We can write 8 as 23 2^3 , because 8 equals 2 multiplied by itself three times.

Let's use the power property of logarithms, which is:

  • logb(an)=nlogba\log_b (a^n) = n \log_b a

Applying this property to log68\log_6 8, we have:

log68=log6(23)\log_6 8 = \log_6 (2^3)

Using the power property, this becomes:

log6(23)=3log62\log_6 (2^3) = 3 \log_6 2

Therefore, the expression for log68\log_6 8 in terms of log62\log_6 2 is:

3log623 \log_6 2.

Answer:

3log62 3\log_62

Video Solution
Exercise #4

xln7= x\ln7=

Step-by-Step Solution

To solve this problem, we'll follow the steps outlined:

  • Step 1: Recognize that the expression xln7 x \ln 7 can be thought of in terms of the power property of logarithms, which helps reframe it into a single logarithm.
  • Step 2: Apply the formula ln(ab)=blna\ln(a^b) = b \ln a. This tells us that if we have something of the form blna b \ln a , we can express it as ln(ab)\ln(a^b).
  • Step 3: Utilize the known expression and rule by substituting a=7 a = 7 and b=x b = x . Thus, xln7 x \ln 7 becomes ln(7x)\ln(7^x).

Therefore, the rewritten expression for xln7 x \ln 7 using logarithm rules is ln7x \ln 7^x .

This matches choice 4 from the provided options.

Answer:

ln7x \ln7^x

Video Solution
Exercise #5

7\log_42<\log_4x

Step-by-Step Solution

To solve the inequality 7log42<log4x7\log_4 2 < \log_4 x, we will follow these steps:

  • Step 1: Simplify 7log427\log_4 2 using the logarithm properties.
  • Step 2: Write the inequality log4(27)<log4x\log_4(2^7) < \log_4 x.
  • Step 3: Solve for xx by converting the logarithmic inequality into an exponential form.

Let's now proceed with these steps:

Step 1: Using the power property of logarithms, we have 7log42=log4(27)7\log_4 2 = \log_4 (2^7). This step simplifies the multiplication into a single logarithmic term.

Step 2: Using substitution in the inequality, we write it as log4(27)<log4x\log_4 (2^7) < \log_4 x.

Step 3: Since logarithms are one-to-one functions, we can conclude that if log4(27)<log4x\log_4 (2^7) < \log_4 x, then 27<x2^7 < x. This results from the property ac=ad    c=da^c = a^d \iff c = d where the bases are equal.

Therefore, the solution to the inequality is 27<x\mathbf{2^7 < x}.

Answer:

2^7 < x

Video Solution

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