Solving equations Solve the following equations.


log₁₀ x= 3

Verified step by step guidance

1. Recognize that the equation \( \log_{10} x = 3 \) is in logarithmic form.
2. Recall the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \).
3. Apply the definition to the given equation: \( \log_{10} x = 3 \) implies \( 10^3 = x \).
4. Calculate \( 10^3 \) to find the value of \( x \).
5. Conclude that \( x \) is the result of \( 10^3 \).

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation. The equation log₁₀ x = 3 means that 10 raised to the power of 3 equals x. Understanding logarithms is essential for solving equations involving them, as they help to express relationships between numbers in a more manageable form.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of the logarithmic equation, recognizing that the logarithm represents an exponent allows us to convert the logarithmic form into an exponential form, facilitating the solution process.

Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, provide tools for simplifying and manipulating logarithmic expressions. While not directly needed for this specific equation, understanding these properties can be crucial for solving more complex logarithmic equations and for combining multiple logarithmic terms.