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Chapter 7: Problem 16

Find the numbers \(x\) which satisfy the equation. $$\log _{x} 2=\log _{3} x$$

Short Answer

Expert verified
There's no real number x that satisfies the equation, since neither \(x=2\) nor \(x=9\) satisfy the original logarithmic equation.

Step by step solution

01

Convert to Exponential Form

By definition of logarithms, the above equation can be rewritten as \(x^{\log _{3} x}=2\) and \(3^{\log _{x} 2} = x\)
02

Simplify

Using the property \(a^{\log _{a} k} = k\), we can simplify the x left side to get \(x=2\), and the right side to get \(x=3^2 = 9\). Now we check both potential solutions.
03

Substitution and Verification

Substitute \(x=2\) into original logarithmic equation, we get \(\log _{2} 2=\log _{3} 2\), with the left side equal to 1 and the right unequal to 1. So \(x=2\) is not a solution. Substitute \(x=9\) into original logarithmic equation, we get \(\log _{9} 2=\log _{3} 9\), with left side unequal to 2 but the right side equal to 2. So \(x=9\) is not a solution either.

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Key Concepts

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

Exponential Form
Understanding logarithms often starts with grasping their exponential form. A logarithm essentially asks the question: "To what power must we raise the base to get a certain number?" For instance, \(\log_{a}b = c\)means that \(a^c = b\). Here, the logarithmic form \(\log_{a}b\) can be converted into exponential form as \(a^c = b\).

In our given problem, \(\log_{x}2 = \log_{3}x\) has two equations that can be rewritten in their exponential forms. The left side becomes \(x^{\log_{3}x} = 2\) and the right side becomes \(3^{\log_{x}2} = x\).Turning logarithmic equations into exponential form can make complex problems simpler and more approachable.
Properties of Logarithms
Logarithms have some amazing properties that can simplify equations. One of these is \(a^{\log_{a}k} = k\). It tells us that if we take the base \(a\) and raise it to the power of the logarithm of \(k\) with that base, the result is simply \(k\). This property is an invaluable tool for simplifying expressions in logarithmic equations.

In our example, for \(x^{\log_{3}x} = 2\), we simplify using the property to find that \(x = 2\). Similarly, for \(3^{\log_{x}2} = x\), we simplify to obtain \(x = 9\).
  • These properties can help to cut down complicated expressions into simpler terms.
  • They are derived from the definition of logarithms and are consistently useful in equations involving logs.
  • Knowing these properties allows us to solve logarithmic equations efficiently.
Substitution Method
The substitution method is a handy tool when verifying solutions in mathematical equations. Once potential solutions are derived from simplifications or transformations, substitution checks their validity.

In our case, we identified two potential values of \(x\): 2 and 9. Simply plug these back into the original logarithmic equation to confirm if they satisfy it.
  • Start with \(x = 2\). The equation becomes \(\log_{2}2 = \log_{3}2\). With \(\log_{2}2 = 1\), observe that the right side doesn't equate, so \(x = 2\) isn't valid.
  • Next, try \(x = 9\). The equation becomes \(\log_{9}2 = \log_{3}9\). Here, the left side doesn't simplify to match the right, thus \(x = 9\) isn't a valid solution either.
Substitution confirms whether a potential solution truly satisfies the equation. It's a fail-safe approach to ensure correctness, particularly useful in complex equations.

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