How to Evaluate Logarithms With Square Root Bases

Math students studying logarithms are familiar with the form log a (b) = x.
••• Jupiterimages/Goodshoot/Getty Images

The logarithm of a number identifies the power that a specific number, referred to as a base, must be raised to produce that number. It is expressed in the general form as log a (b) = x, where a is the base, x is the power that the base is being raised to, and b is the value in which the logarithm is being calculated. Based on these definitions, the logarithm can also be written in exponential form of the type a^x=b. Using this property, the logarithm of any number with a real number as the base, such as a square root, can be found following a few simple steps.

    Convert the given logarithm to exponential form. For example, the log sqrt(2) (12) = x would be expressed in exponential form as sqrt(2)^x = 12.

    Take the natural logarithm, or logarithm with base 10, of both sides of the newly formed exponential equation.
    log( sqrt(2)^x) = log (12)

    Using one of the properties of logarithms, move the exponent variable to the front of the equation. Any exponential logarithm of the type log a ( b^x) with a particular "base a" can be rewritten as x_log a (b). This property will remove the unknown variable from the exponent positions, thereby making the problem much easier to solve. In the previous example, the equation would now be written as: x_log(sqrt(2)) = log (12)

    Solve for the unknown variable. Divide each side by the log(sqrt(2)) to solve for x: x=log(12)/log(sqrt(2))

    Plug this expression into a scientific calculator to get the final answer. Using a calculator to solve the example problem gives the final result as x = 7.2 .

    Check the answer by raising the base value to the newly calculated exponential value. The sqrt(2) raised to a power of 7.2 results in the original value of 11.9, or 12. Therefore, the calculation was done correctly:

    sqrt(2)^7.2 =11.9

Related Articles

How to Convert Log Scale to Linear
How to Solve Logarithms With Different Bases
How to Calculate Antilog
How to Get Rid of Cubed Power
How to Write in Exponential Form
How to Put Base Log on Graphing Calculator
How to Use Log on a TI-83
How to Enter a Subscript on the TI-83
How to Do Powers in Math
How to Convert Exponents to Logs
How to Calculate Log2
How to Get Rid of Logarithms
What Does YX Mean on a Calculator?
How to Convert Log Scale to Linear
Sulfuric Acid & Chlorine Bleach Reaction
How to Calculate the Volume of a Cylinder
How to Divide Using Logarithms
How to Use the Ti84 Calculator to Add Logs
What Is a Mathematical Expression?
How to Solve Equations for the Indicated Variable

Dont Go!

We Have More Great Sciencing Articles!