/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Simplify the expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 60

Simplify the expression. $$ \log _{6} 6^{7} $$

Short Answer

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Step by step solution

01

Understand the logarithm property

Recall the logarithm power rule: \( \log_b (a^c) = c \log_b (a)\). This property will help in simplifying the given expression.
02

Apply the logarithm power rule

Here, \log_{6} (6^7)\ can be simplified using the property: \( \log_b (a^c) = c \log_b (a)\). Applying this, we get: \( \log_{6} (6^7) = 7 \log_{6} (6)\).
03

Simplify \log_b (b)\

For any base \( b \), \( \log_b (b) = 1\). Therefore, \( \log_6 (6) = 1\).
04

Final simplification

Using the result from Step 3, simplify \( \log_{6} (6^7) = 7 \log_{6} (6)\) to get: \( 7 \times 1 = 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm power rule
The logarithm power rule is a crucial property in simplifying logarithmic expressions. It states that for any positive numbers a and b, and positive integer c:
\( \log_b (a^c) = c \log_b (a) \)
This means you can take the exponent of the argument (a) and multiply it by the logarithm of the base (b).
Essentially, it allows you to 'move' the exponent in front as a coefficient, simplifying the expression significantly.
simplification
Simplification involves reducing an expression to its simplest form.
For logarithmic expressions, using properties such as the logarithm power rule can turn complex expressions into simpler ones.
In the given exercise, we started with the expression \( \log_{6} (6^7) \).
By applying the logarithm power rule, we were able to simplify it to \( 7 \log_{6} (6) \), which further simplifies to 7, because \( \log_{6} (6) = 1 \).
The step-by-step breakdown allows us to understand each part of the process, making it easier to handle similar problems in the future.
logarithm base rule
The logarithm base rule states that the logarithm of a number to its own base is always 1.
Mathematically, this is expressed as \( \log_b (b) = 1 \).
This rule simplifies many logarithmic expressions by reducing any \( \log_b (b) \) term to 1.
In our exercise, we used this rule to simplify \( \log_6 (6) \) to 1, which then allowed us to further simplify \( 7 \log_{6} (6) \) to 7.
exponents
Exponents are a way of expressing repeated multiplication of the same number.
For example, \( 6^7 \) means 6 multiplied by itself 7 times.
In the context of logarithms, exponents play a key role, especially when using the power rule.
The power rule essentially deals with transforming the exponent into a coefficient, aiding in simplification.
Remembering these properties and rules helps in efficiently tackling logarithmic expressions and understanding their underlying mechanics.

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Most popular questions from this chapter

Show that \(-\ln \left(x-\sqrt{x^{2}-1}\right)=\ln \left(x+\sqrt{x^{2}-1}\right)\).

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{3} 15 $$

Refer to the model \(Q(t)=Q_{0} e^{-0.000121 t}\) used in Example 5 for radiocarbon dating. At the "Marmes Man" archeological site in southeastern Washington State, scientist uncovered the oldest human remains yet to be found in Washington State. A sample from a human bone taken from the site showed that \(29.4 \%\) of the carbon-14 still remained. How old is the sample? Round to the nearest year.

Solve for the indicated variable. \(N=N_{0} e^{-0.025 t}\) for \(t\) (used in chemistry)

Painful bone metastases are common in advanced prostate cancer. Physicians often order treatment with strontium- \(89\left({ }^{89} \mathrm{Sr}\right)\), a radionuclide with a strong affinity for bone tissue. A patient is given a sample containing \(4 \mathrm{mCi}\) of \({ }^{89} \mathrm{Sr}\). a. If \(20 \%\) of the \({ }^{89} \mathrm{Sr}\) remains in the body after 90 days, write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the amount \(Q(t)\) of radioactivity in the body \(t\) days after the initial dose. b. What is the biological half-life of \({ }^{89} \mathrm{Sr}\) under this treatment?
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