Problem 10 - Elementary and Intermediate Algebra

Question

Fill in the blanks. $$ \log x=-2 \text { is equivalent to }= $$

Step-by-Step Solution

Answer

\( \log x = -2 \) is equivalent to \( x = 0.01 \).

Step 1: Understanding Logarithmic Forms

The equation given is in the logarithmic form. The equation is \( \log x = -2 \). This can be understood in the logarithmic identity:\[ \log_b a = c \] is equivalent to \[ a = b^c \].Here, the common logarithm \( \log x \) means \( \log_{10} x \), hence \( b = 10 \).

Step 2: Applying the Logarithmic Identity

Using the logarithmic identity recognized in Step 1, convert the equation \( \log_{10} x = -2 \) into its equivalent exponential form. The identity tells us that if \( \log_b a = c \), then it follows that \( a = b^c \). Hence, here:\[ x = 10^{-2} \].

Step 3: Calculating the Exponential Form

Now, calculate the value of \( 10^{-2} \). Recall that any number raised to a negative exponent is the reciprocal of that number raised to the corresponding positive exponent:\[ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01 \].

Key Concepts

Exponential Form

The exponential form is an alternative way to write equations that involve exponents. If you have a logarithmic equation like \( \log_b a = c \), it can be rewritten in exponential form as \( a = b^c \). This transformation allows us to view the problem from another angle, making it easier for many to understand.In the given exercise, we start with \( \log_{10} x = -2 \). By applying the exponential form, we convert it to \( x = 10^{-2} \). Here, "b" is the base of the logarithm, which is 10 for common logarithms. This approach helps to simplify solving the problem, as one can directly work with exponential expressions.

Negative Exponents

Negative exponents might seem daunting at first, but they represent a simple idea: the reciprocal of the number raised to the positive power. For example, if you have \( 10^{-2} \), it means \( \frac{1}{10^2} \).Understanding this can be crucial for solving problems involving logarithms and exponents. Instead of thinking of the expression as shrinking, consider it as flipping or inverting the base. Calculating \( 10^{-2} \), we find the reciprocal of \( 10^2 \), which is 100. Hence, \( \frac{1}{100} = 0.01 \).

• \( 10^2 = 100 \)
• \( 10^{-2} = \frac{1}{100} \)
• As a decimal, this results in 0.01

Understanding negative exponents aids in navigating between logarithmic and exponential forms efficiently.

Common Logarithms

Common logarithms are logarithms with base 10. They are widely used in mathematical problems and practical applications, especially when dealing with concepts like pH in chemistry or decibels in sound measurement.In expressions like \( \log x \), the absence of a base indicates a common logarithm, implying the base is 10. This standard practice helps in simplifying expressions and calculations where base conversion is unnecessary.Using common logarithms might simplify a problem since calculators and various mathematical tools are often set to handle these by default. Recognizing and converting these forms, as in our original problem where \( \log_{10} x = -2 \), enables an effective solution path through exponential and negative exponent concepts.