Contextual Applications of Derivatives: Background You’ll Need 1

  • 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:

logb(x)=yby=x,b>0,b1

Here, b is always a positive number and cannot be equal to 1.

Think b to the y equals x.

The logarithm function logb(x) is conventionally written with parentheses to clearly denote the function’s input, similar to f(x). However, when dealing with a single variable or a simple expression, parentheses might be omitted, leading to the notation logbx. It’s important to note that many calculators might still require parentheses around the input x.

The notation logb(c)=a can be interpreted as ba=c. This implies that the base b raised to the power a equals c.

logb (c) = a means b to the A power equals C.

It’s common to see ln representing the natural logarithm, which uses e (approximately 2.718) as its base. The notation ln corresponds to loge, emphasizing the natural logarithm’s specific base.

logarithmic functions

A logarithm base b of a positive number x satisfies the following definition: For x>0,b>0,b1, y=logb(x) is equal to by=x, where

  • we read logb(x) as, “the logarithm with base b of x” or the “log base b of x.”
  • the logarithm y is the exponent to which b must be raised to get x.
  • if no base b is indicated, the base of the logarithm is assumed to be 10.

Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b is(0,).
  • the range of the logarithm function with base b is(,).

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 logb(x)=y:

  1. Identify the Components: Recognize b as the base, y as the logarithmic result, and x as the argument of the logarithm.
  2. Convert to Exponential Form: Rewrite the equation from logarithmic to exponential form by setting the base b raised to the power y equal to x. This translates to by=x.

Write the following logarithmic equations in exponential form.

  1. log6(6)=12
  2. log3(9)=2
  3. log10(1,000,000)=6
  4. log5(25)=2


Convert from Exponential to Logarithmic Form

To convert from exponential to logarithmic form, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write x=logb(y).

Write the following exponential equations in logarithmic form.

  1. 23=8
  2. 52=25
  3. 104=110,000