Question

Evaluate the logarithm. Round your result to three decimal places.\(\log _{1 / 4} 10\)

Short answer

Evaluating the value of x in the calculator provides x as approximately -1.660.

Step-by-step solution

Understanding Logarithms

Asking \( \log _{1 /4} 10 \) is equivalent to asking what power you need to raise \( \frac{1}{4} \) to, in order to get 10. Logarithms are the reverse operation of exponentiation.

Translate the log function into exponential form

Translate the logarithmic equation into an equivalent exponential equation using the basic property that \A^n=B is equivalent to \log_A B = n. In this case, \( \frac{1}{4} ^x = 10 \), where x is the value we are looking for. However, this is not easily solvable as is.

Use the property of logarithms

Using the properties of logarithms, we can rewrite the base \( \frac{1}{4} \) as \( 2^{-2} \). This gives us a new equation which is \( 2^{-2x} = 10 \).

Convert to Natural Log

To work with this further, the equation could be rewritten by taking the natural log on both sides, this gives us \(-2x \ln(2)= \ln(10) \) .

Solve for x

Continue to rearrange and solve for x by dividing both sides by \(-2 \ln(2)\) resulting in \( x = -\frac{\ln(10)}{2 \ln(2)} \) . . This value can be evaluated.

Key concepts

Logarithmic to Exponential Form

Understanding how to convert logarithmic statements to exponential form is crucial in mathematics, especially when evaluating less intuitive logarithmic expressions such as \( \log _{1 /4} 10 \).

To do this, remember that the log equation \( \log_{b} a = c \) is equivalent to the exponential equation \( b^c = a \). Here, the base \( b \) of the logarithm becomes the base of the exponent, the result \( c \) of the log is the exponent, and the number \( a \) that we're taking the log of is the result of the exponentiation. When applied to the given problem, we translate \( \log _{1 /4} 10 \) into \( (\frac{1}{4})^x = 10 \), where \( x \) represents the power that \( \frac{1}{4} \) is raised to reach 10. It can be tough to solve this directly, but with the use of logarithmic properties, we can simplify the equation and solve it with greater ease.

Properties of Logarithms

The properties of logarithms are essential tools for rewriting complex logarithmic expressions into more manageable forms. Some key properties include the Product Rule, Quotient Rule, and the Power Rule.

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \), states that the log of a product is equal to the sum of the logs of its factors.
• Quotient Rule: \( \log_b (\frac{m}{n}) = \log_b m - \log_b n \), implies that the log of a quotient is the difference of the logs of the numerator and the denominator.
• Power Rule: \( \log_b (m^n) = n\log_b m \), allows us to move the exponent of the argument outside in front of the log.

By applying these properties, complex equations like \( 2^{-2x} = 10 \) become easier to solve, as they can often be converted into a form that's solvable through basic algebraic manipulation.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm that uses the mathematical constant \( e \) as its base. It is widely used in mathematics, physics, and engineering. The value of \( e \) is approximately 2.718, and it is the base rate of growth shared by all continually growing processes.

When dealing with challenging logarithms like \( \log_{1/4} 10 \), converting to natural logarithms can be very useful. For example, converting an equation like \( 2^{-2x} = 10 \) into natural logarithmic form allows us to use the properties of logarithms in tandem with the convenient derivative and integral properties of the natural logarithm. This gives us \( -2x \ln(2)= \ln(10) \), enabling the use of algebraic techniques to solve for \( x \).

Solving Logarithmic Equations

To solve logarithmic equations, you'll typically follow a sequence of converting the log equation to its exponential form, applying logarithmic properties for simplification and then solving the resulting equation using algebra.

For our example, once we have the natural log equation \( -2x \ln(2)= \ln(10) \), we can solve for \( x \) by isolating it on one side of the equation. We do this by dividing both sides by (-2 \ln(2)) getting us to \( x = -\frac{\ln(10)}{2 \ln(2)} \). When you reach this step, you can often use a calculator to evaluate the expression and round it to the desired decimal places as needed. This method of solving ensures that students can approach even the most complex logarithmic equations with a structured strategy.