Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation. $$\log _{10} x=-\frac{1}{4}$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation is \(x = \frac{1}{\sqrt[4]{10}}\)

Step 1: Rewrite the equation in exponential form

In general, a base (b), an exponent (y), and a result (x) are related in the following way: \(b^y = x\). Hence the given equation \(\log_{10} x=-\frac{1}{4}\) can be rewritten in exponential form as \(10^{-\frac{1}{4}} = x\)

Step 2: Calculate the value of x

To calculate \(10^{-\frac{1}{4}}\), remember that a negative exponent indicates a reciprocal, and a fractional exponent is the same as a root, so \(10^{-\frac{1}{4}} = \frac{1}{10^\frac{1}{4}}\). It's still base 10, so let's convert it into a root, such that \(10^\frac{1}{4}\) is equal to the fourth root of 10. Thus, \(x = \frac{1}{\sqrt[4]{10}}\)

Key Concepts

Exponential Form

When dealing with logarithmic equations like \( \log_{10} x = -\frac{1}{4} \), it is often helpful to convert them into exponential form.
Exponential form is a mathematical expression where numbers are written as a base raised to an exponent. Essentially, it’s the reverse of a logarithm. In general terms, the relationship can be expressed as \( b^y = x \), where \( b \) is the base, \( y \) the exponent, and \( x \) is the result.

This conversion simplifies calculations by allowing you to see the equation in terms of multiplication instead of a function of a logarithm. For example, in our problem, converting \( \log_{10} x = -\frac{1}{4} \) into exponential form gives us \( 10^{-\frac{1}{4}} = x \). This equation now explicitly shows \( x \) as a power of 10, making it straightforward to solve using rules of exponents.

Negative Exponents

Negative exponents can be a tricky concept to grasp, but they are actually quite straightforward once you understand the basic rule: any non-zero number raised to a negative exponent is the reciprocal of that number raised to the opposite positive exponent.

• For instance, \( a^{-n} = \frac{1}{a^n} \).

This rule is very useful when solving equations in exponential form, particularly when the exponent is negative.

In our example, \( 10^{-\frac{1}{4}} \) can be re-written as \( \frac{1}{10^{\frac{1}{4}}} \). By doing so, we convert a potentially confusing expression into a much simpler, manageable fraction. This helps identify that the result, \( x \), is actually the reciprocal of \( 10^{\frac{1}{4}} \), rather than a direct power of 10.

Fractional Exponents

Fractional exponents are another essential aspect when dealing with equations in exponential form. A fractional exponent denotes both a power and a root. Consider the general structure \( a^{m/n} \), where \( m \) is the power and \( n \) is the root.

• This can be interpreted as the \( n \)-th root of \( a \), raised to the \( m \)-th power: \( (\sqrt[n]{a})^m \).
• Alternatively, \( a^{m/n} = (a^m)^{1/n} \).

Returning to our example, \( 10^{1/4} \) specifically represents the fourth root of 10. By assigning 1 as the power and 4 as the root, we can also express this as \( \sqrt[4]{10} \), making it easier to evaluate and visualize in simpler mathematical terms. This understanding is crucial to solving equations and working with complex expressions smoothly and effectively.

Roots

Roots are the inverse operation of exponents. While an exponent raises a base to a power, a root finds what base was raised to that power. Roots can be of various degrees, such as square roots, cube roots, and in our case, fourth roots.

• The square root of a number \( a \) is symbolized as \( \sqrt{a} \).
• For a cube root, it's \( \sqrt[3]{a} \).
• Correspondingly, a fourth root is written as \( \sqrt[4]{a} \).

In our example, calculating \( \sqrt[4]{10} \) becomes significant because it converts the fractional exponent \( 10^{\frac{1}{4}} \) into a more tangible expression. When finding roots, you essentially determine the number that, when used in a repeated multiplication process, results in the original value. Being proficient in handling roots helps simplify equations and is crucial in understanding solutions involving fractional exponents.