Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log _{y}(x)=-11\)

Step-by-Step Solution

Answer

The exponential form is \(y^{-11} = x\).

Step 1: Understanding the Logarithmic Form

The given equation is in the logarithmic form: \(\log_{y}(x) = -11\). This means that \(y\) raised to what power equals \(x\)? That's the definition of the logarithmic equation; finding the exponent "-11" to which the base \(y\) must be raised to obtain \(x\).

Step 2: Rewriting in Exponential Form

To convert the logarithmic equation \(\log_{y}(x) = -11\) into exponential form, use the conversion rule: if \(\log_{b}(a) = c\), then \(b^{c} = a\). Apply this to the equation: \(y^{-11} = x\).

Step 3: Conclusion of the Conversion

By following the conversion rule, the logarithmic equation \(\log_{y}(x) = -11\) is rewritten in exponential form as \(y^{-11} = x\). This indicates that \(x\) is the result of raising \(y\) to the power of \(-11\).

Key Concepts

Logarithmic Form

Logarithmic form is a way of expressing an equation where a base, an exponent, and a result are involved. Typically, it answers the question: 'To what power should the base be raised to obtain the result?' For example, in the expression \(\log_{y}(x) = -11\), \(y\) is the base, and \(-11\) is the exponent, implying that \(y\) must be raised to this power of \(-11\) to yield \(x\). This form is important for simplifying and solving problems involving exponential relationships in mathematics.

Logarithms are particularly useful because they help convert multiplicative relationships into additive ones. This can make complex calculations easier to handle. When you see a logarithmic expression, you're looking at a compact way of representing repeated multiplication.

Converting Equations

Converting equations from one form to another is a valuable skill. It can make equations easier to understand and solve. In our context, we are converting between logarithmic and exponential forms.

To convert a logarithmic equation like \(\log_{b}(a) = c\) into its exponential form, apply the rule: \(b^{c} = a\). This means that if the logarithm of \(a\) to the base \(b\) is \(c\), then \(a\) is \(b\) raised to the power of \(c\).

In practice, this is helpful because it allows you to switch between forms depending on what is more convenient for solving the problem. When you have a logarithmic equation and need an exponential form, simply apply this conversion rule to find your answer.

Exponents

Exponents are a way to denote repeated multiplication. For example, \(b^{c}\) means you multiply \(b\) by itself \(c\) times. They are fundamental in algebra and are widely used across different domains of mathematics.

Understanding how exponents work is critical. With positive exponents, you multiply the base by itself. However, with negative exponents, like \(y^{-11}\), you take the reciprocal of the base raised to the positive exponent. For instance, \(y^{-11} = \frac{1}{y^{11}}\).

Negative exponents reflect how small values can become when dividing rather than multiplying. Mastering how both positive and negative exponents modify numbers can greatly enhance problem-solving abilities in algebra and beyond.