Convert between exponential and logarithmic form: all bases

Learning Objectives Define the inverse relationship between exponential and logarithmic functions. Identify the base, argument, and exponent in both exponential and logarithmic equations. Convert an exponential equation of any valid base into its equivalent logarithmic form. Convert a logarithmic equation of any valid base into its equivalent exponential form. Evaluate simple logarithmic expressions by first converting them to exponential form. State the constraints on the base and argument of a logarithm. Solve for a single unknown variable in simple logarithmic equations by converting to exponential form. Ever wondered how scientists measure earthquake intensity or sound levels? They use a mathematical 'secret code' called logarithms to handle huge numbers easi...

TermDefinitionExample Exponential FormAn equation written in the form b^y = x, where a base 'b' is raised to an exponent 'y' to get a result 'x'.2^5 = 32 Logarithmic FormAn equation written in the form log_b(x) = y, which asks the question: 'To what exponent must the base 'b' be raised to get the argument 'x'?'log_2(32) = 5 Base (b)The number being raised to a power in exponential form, or the subscript number in logarithmic form. The base must be positive and not equal to 1 (b > 0, b ≠ 1).In both 10^2 = 100 and log_10(100) = 2, the base is 10. Argument (x)The result of the exponential expression, or the value inside the logarithm. The argument must be positive (x > 0).In both 3^4 = 81 and log_3(81) = 4, the argument is 81....

The Fundamental Conversion Rule b^y = x <=> log_b(x) = y This is the core relationship for converting between the two forms. The base of the exponent becomes the base of the logarithm. The exponent is what the logarithm equals. Base and Argument Constraints For log_b(x) to be a real number: b > 0, b ≠ 1, and x > 0 These conditions must be met for a logarithm to be well-defined in the real number system. You cannot take the log of a negative number or zero, and the base cannot be negative, zero, or one.