- Change logarithmic equations into exponential equations
Convert from Logarithmic to Exponential Form
Understanding the relationship between logarithmic and exponential forms is fundamental. This conversion can be succinctly represented as follows:
[latex]{\mathrm{log}}_{b}\left(x\right)=y\Leftrightarrow {b}^{y}=x,\text{}b>0,b\ne 1[/latex]
Here, [latex]b[/latex] is always a positive number and cannot be equal to [latex]1[/latex].
The logarithm function [latex]\log_{b}(x)[/latex] is conventionally written with parentheses to clearly denote the function’s input, similar to [latex]f(x)[/latex]. However, when dealing with a single variable or a simple expression, parentheses might be omitted, leading to the notation [latex]\log_{b}x[/latex]. It’s important to note that many calculators might still require parentheses around the input [latex]x[/latex].
The notation [latex]\log_{b}(c)=a[/latex] can be interpreted as [latex]b^a=c[/latex]. This implies that the base [latex]b[/latex] raised to the power [latex]a[/latex] equals [latex]c[/latex].
It’s common to see [latex]ln[/latex] representing the natural logarithm, which uses [latex]e[/latex] (approximately [latex]2.718[/latex]) as its base. The notation [latex]ln[/latex] corresponds to [latex]\log_e[/latex], emphasizing the natural logarithm’s specific base.
logarithmic functions
A logarithm base [latex]b[/latex] of a positive number [latex]x[/latex] satisfies the following definition: For [latex]x>0,b>0,b\ne 1[/latex], [latex]y={\mathrm{log}}_{b}\left(x\right)\text{ is equal to }{b}^{y}=x[/latex], where
- we read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, “the logarithm with base [latex]b[/latex] of [latex]x[/latex]” or the “log base [latex]b[/latex] of [latex]x[/latex].”
- the logarithm y is the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex].
- if no base [latex]b[/latex] is indicated, the base of the logarithm is assumed to be [latex]10[/latex].
Also, since the logarithmic and exponential functions switch the [latex]x[/latex] and [latex]y[/latex] values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
- the domain of the logarithm function with base [latex]b \text{ is} \left(0,\infty \right)[/latex].
- the range of the logarithm function with base [latex]b \text{ is} \left(-\infty ,\infty \right)[/latex].
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
How To: Convert a Logarithmic Equation to Exponential Form
Given a logarithmic equation in the format [latex]\log_{b}(x)=y[/latex]:
- Identify the Components: Recognize [latex]b[/latex] as the base, [latex]y[/latex] as the logarithmic result, and [latex]x[/latex] as the argument of the logarithm.
- Convert to Exponential Form: Rewrite the equation from logarithmic to exponential form by setting the base [latex]b[/latex] raised to the power [latex]y[/latex] equal to [latex]x[/latex]. This translates to [latex]b^y=x[/latex].
Write the following logarithmic equations in exponential form.
- [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex]
- [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
- [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex]
- [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex]
Convert from Exponential to Logarithmic Form
To convert from exponential to logarithmic form, we follow the same steps in reverse. We identify the base [latex]b[/latex], exponent [latex]x[/latex], and output [latex]y[/latex]. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].
Write the following exponential equations in logarithmic form.
- [latex]{2}^{3}=8[/latex]
- [latex]{5}^{2}=25[/latex]
- [latex]{10}^{-4}=\frac{1}{10,000}[/latex]