Question

Solve each equation. $$ \log _{2} 8^{x}=-6 $$

Short answer

x = -2

Step-by-step solution

Recall the properties of logarithms

Remember that logarithms have properties that can simplify expressions. One such property is \(\log_b(a^c) = c \log_b(a)\). This will help to simplify the given equation.

Apply the property to the given equation

We begin by applying the property \(\log_b(a^c) = c \log_b(a)\) to the equation \(\log_{2} 8^{x} = -6\). This simplifies to \(x \log_{2} 8 = -6\).

Calculate the logarithm

Next, determine the value of \(\log_{2} 8\). Since \(8 = 2^3\), we have \(\log_{2} 8 = \log_{2} 2^3 = 3\). This gives us the equation \(x \cdot 3 = -6\).

Solve for x

Now, solve the equation \(3x = -6\) by dividing both sides by 3: \(x = \frac{-6}{3} = -2\).

Key concepts

logarithm properties

Understanding the properties of logarithms is key to solving logarithmic equations. One essential property is the power rule of logarithms, which states that \( \log_b(a^c) = c \log_b(a) \ \). This rule allows us to bring the exponent in a logarithmic expression to the front. Another important property is the change of base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \ \). These properties help us simplify and solve logarithmic equations effectively.

solving equations

To solve logarithmic equations, we use logarithm properties to simplify. Let’s illustrate with our example. Given the equation \( \log_{2} 8^{x} = -6 \ \), we start by applying the logarithm power rule \( \log_b(a^c) = c \log_b(a) \ \). This transforms the given equation into \ x \log_{2} 8 = -6 \. Next, we calculate \ \log_{2} 8 \, knowing that 8 can be expressed as \ 2^3 \:. \ \log_{2} 8 = \log_{2} 2^3 = 3 \. Substituting back, we get \ 3x = -6 \. Finally, solve for \ x \ by dividing both sides by 3, resulting in \ x = -2 \$.

change of base formula

The change of base formula \(\frac{\log_c(a)}{\log_c(b)} \) is a handy tool when dealing with logarithms of different bases. If you need to evaluate a logarithm in a base that your calculator doesn't support directly, you can convert it using the change of base formula. For instance, to find \ \log_2 8 \, if your calculator only has base-10 or natural log (ln), you can use \ \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \ or \ \log_2 8 = \frac{\ln 8}{\ln 2} \. This formula ensures flexibility and can assist in various logarithmic calculations.