The function f(x) = logₐx is a logarithmic function with the base of 'a'. This log function is the inverse of the exponential function.
Logarithmic Functions Examples
For example, log₁8 = 3, because logₐb = c asks "what do you raise 'a' to get 'b'?", so in this case, log₁8 = 3, because 1 raised to the power of 3 equals 8.
Log Rules
The logarithmic function can be rewritten in the form of an exponential function by using the log rules, as shown in the following examples.
Definition of Logarithm in Mathematics
When the base is 10, f(x) = log x is a common logarithm. So, log 1000 = log₁₀ 1000 = 3, as 10 raised to the power of 3 equals 1000.
When the base is 'e', f(x) = ln x is a natural logarithm with 'e' being approximately 2.71828. For example, loge 5.36 = 1.68.
Simple Logarithmic Functions Examples and Calculator
We can use a simple logarithmic functions calculator to solve logarithmic equations - for instance, logₐ(w) = 2 can be solved to get w = 81. Similarly, ln x = n can be solved to get x = eⁿ, and log 10x = x can be solved to get x ≈ 0.401.
Rewriting Logarithmic Functions
Logarithmic functions can be rewritten in exponential form. For example, when f(x) = 10(1.75)^x, it can be rewritten as f(x) = a * b^x, where a = 10 and b ≈ 1.175.
Logarithm Examples with Solutions
Another example is log₃(x) = 8, which can be solved to find that x = 3^8. You can also solve exponential equations like 8^(3x-2) = 107 to find the value of x using logarithms.
Solving logarithmic equations may involve converting them to exponential form and using log calculators. You can also convert exponential equations to logarithmic form to solve them. Logarithmic equations with different bases can be solved step-by-step as demonstrated in the examples above.
By using log rules, log to exponential form worksheet, and log to exponential form examples, one can understand the properties of log and exponential functions as inverses of each other. This understanding can be applied to solve a variety of logarithmic equations.
In conclusion, understanding logarithmic functions, using log calculators, and practicing with simple logarithmic functions worksheets can improve your ability to solve logarithmic equations.
