In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874

Verified step by step guidance

1. Identify the general relationship between exponential and logarithmic forms. The exponential form is written as a^b = c, and its equivalent logarithmic form is log_a(c) = b.
2. Compare the given equation 13^y = 874 with the general exponential form a^b = c. Here, the base (a) is 13, the exponent (b) is y, and the result (c) is 874.
3. Rewrite the equation in logarithmic form using the relationship log_a(c) = b. Substitute the base (13), the result (874), and the exponent (y) into the logarithmic form.
4. The equivalent logarithmic form of the equation is log_13(874) = y.
5. This means that y is the power to which 13 must be raised to produce 874.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. In the equation 13^y = 874, 13 is the base, y is the exponent, and 874 is the result of the exponentiation. Understanding how to manipulate these equations is crucial for converting them into logarithmic form.

Logarithmic Form

Logarithmic form is a way to express exponential equations using logarithms. The equation a^b = c can be rewritten as log_a(c) = b, where 'a' is the base, 'c' is the result, and 'b' is the exponent. This transformation is essential for solving equations involving exponents and understanding their properties.

Properties of Logarithms

Properties of logarithms include rules that simplify the manipulation of logarithmic expressions, such as the product, quotient, and power rules. These properties help in solving logarithmic equations and understanding their behavior. Familiarity with these rules is important when converting exponential equations to logarithmic form.