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Chapter 12: Problem 87

Solve. If no solution exists, state this. $$ \log _{x}\left(\log _{3} 27\right)=3 $$

Short Answer

Expert verified
The solution is \( x = \sqrt[3]{3} \).

Step by step solution

01

Simplify the Inner Logarithm

First, simplify the inner logarithm, \(\text{log}_{3} 27\). Recall that \27\ is a power of 3, specifically \27 = 3^3\. Therefore, \ \text{log}_{3} 27 = 3 \ because \3^3 = 27\.
02

Substitute the Simplified Logarithm

Substitute \( \text{log}_{3} 27 = 3 \) into the original equation: \( \text{log}_{x} 3 = 3 \). The equation now reads: \( \text{log}_{x} 3 = 3 \).
03

Convert to Exponential Form

Convert the logarithmic equation to its exponential form. The equation \( \text{log}_{x} 3 = 3 \) can be written as: \( x^3 = 3 \).
04

Solve for the Base x

Solve for \( x \) by taking the cube root of both sides of the equation: \( x = \sqrt[3]{3} \).

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Key Concepts

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

Solve Logarithms
When solving logarithmic equations, you are often looking to simplify and solve for the unknown. Start by examining the structure of the logarithmic equation. Break down complicated expressions step by step.

In this original exercise, we began with the equation \(\text{log}_{x}(\text{log}_{3} 27)=3\). We recognized that we needed to simplify the inner logarithm first.
Here are some steps to help in solving logarithmic equations:
  • Identify any inner logarithmic expressions.
  • Use known mathematical properties, such as \(\text{log base a of a power}\).
  • Substitute the simplified values back into the equation.
By following these steps, we simplified \(\text{log}_{3} 27\) to 3, making the equation easier to handle.
Exponential Form
Converting a logarithmic equation to its exponential form can make it much easier to solve. Remember that logarithms are just another way to express exponents. The exponential form can make the relationships more straightforward.

In our example, after simplifying the inner logarithm, we ended with the equation \(\text{log}_{x} 3 = 3\). The next step was to convert this into exponential form.
Here are steps you can follow:
  • Rewrite \(\text{log}_{b} a \) as \ a = b^c\.
  • Identify the base (\b\bl), the result (\b\bl), and the power (\b\bl).
In our case, we converted \(\text{log}_{x} 3 = 3\) into the exponential form: \ x^3 = 3\. This form often makes it easier to solve for the variable, as it turns the logarithmic relationship into a more direct algebraic equation.
Properties of Logarithms
Understanding the properties of logarithms is incredibly important for solving logarithmic equations efficiently. These properties allow you to simplify complex logarithmic expressions.

Let鈥檚 look at some essential properties:
  • The logarithm of a product: \ \text{log}_{b}(xy) = \text{log}_{b} x + \text{log}_{b} y \.
  • The logarithm of a quotient: \ \text{log}_{b}(x/y) = \text{log}_{b} x - \text{log}_{b} y\.
  • The logarithm of an exponent: \ \text{log}_{b}(x^n) = n \text{log}_{b} x\.

In our exercise, we used the property that \text(log base a of a power equals the exponent)\ to simplify \(\text{log}_{3} 27\). By recognizing that \ 27 = 3^3 \,\ we utilized this property to simplify it to 3. Mastering these properties will help you tackle any logarithmic equation with confidence.

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Most popular questions from this chapter

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}(2 x+10)-\log _{a}\left(x^{2}-25\right)$$

Find the domain of each function. $$g(x)=\sqrt{10-x}$$

$$\text { If } \log _{a} x=2, \text { what is } \log _{a}(1 / x) ?$$

For each function given below, (a) determine the domain and the range, (b) set an appropriate window, and (c) draw the graph. $$ f(x)=2 x^{3} \ln x $$

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