Question

Find the value of \(x\) in each expression. $$x=\log _{27} 9$$

Short answer

\(x = \frac{2}{3}\).

Step-by-step solution

Understand the Logarithm

To solve for x in the expression \(x=\log_{27} 9\), remember that the logarithm equation \(y=\log_{b} a\) can be rewritten in exponential form as \(b^y = a\).

Convert to Exponential Form

Convert the logarithmic equation \(x=\log_{27} 9\) to its exponential form to solve for x, which gives us \(27^{x} = 9\).

Simplify the Equation

Notice that 27 and 9 are both powers of 3; 27 is \(3^3\) and 9 is \(3^2\). So, we can rewrite the equation as \((3^3)^x = 3^2\).

Apply the Power to Power Rule

Using the power to power rule, we multiply the exponents and obtain \(3^{3x} = 3^2\).

Set the Exponents Equal to Each Other

Since the bases are equal and nonzero, set the exponents equal to each other to solve for x: \(3x = 2\).

Solve for x

Divide both sides of the equation by 3 to isolate x, which gives us \(x = \frac{2}{3}\).

Key concepts

Logarithmic Equations

Logarithmic equations are equations that contain logarithms within them. They often require you to find an unknown variable for which the equation holds true. Understanding how to navigate these equations is essential for many mathematical and real-world applications.

To begin solving a logarithmic equation like \(x=\log_{27} 9\), first remember that a logarithm is asking the question: to what power do we need to raise the base, in this case, 27, to get the number 9? This setup is a logarithmic form that can be converted to its exponential counterpart to simplify the problem-solving process.

Many students encounter difficulties with logarithmic equations because they don't recognize the relationship between logs and exponents. Crafting a bridge between these concepts by rewriting logarithms in exponential form is often the breakthrough strategy that illuminates the path to the solution.

Exponential Form

Exponential form is a powerful tool to unravel logarithmic equations. In the given problem, we have \(x = \log_{27} 9\). To convert from its logarithmic form to exponential form, wrangle the components of the logarithmic equation into an equation where the base (27 in this case) is raised to the unknown power (x) to yield the number (9), resulting in \(27^x = 9\).

Exponential form lays bare the fundamental connection between the base, its exponent, and the result. By reconfiguring a log equation into this form, students are often better able to visualize and thus solve the problem, mainly because the operation of exponentiation is a direct contrast to logarithms and tends to be more intuitive.

Power to Power Rule

The power to power rule is a crucial exponent rule that is used when an exponential expression is raised to another power, as is often seen when simplifying logarithmic equations. According to the rule, when you have a power raised to another power, like \( (a^b)^c \), you multiply the exponents: \( a^{b \times c} \).

In our problem, we apply this rule to \( (3^3)^x = 3^2 \). By multiplying the exponents, we get \(3^{3x} = 3^2\). This simple yet powerful property allows us to compare the exponents directly because the bases are now the same. Recognizing when and how to apply the power to power rule often streamlines complex problems and leads students to a clearer understanding and quicker solutions.