# Lesson Video: Converting between Logarithmic and Exponential Forms Mathematics

In this video, we will learn how to convert between logarithmic and exponential forms of equations.

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### Video Transcript

In this video, we will learn how to convert equations between logarithmic and exponential form. We will begin by identifying what an equation looks like in both of these forms.

A logarithmic function is the inverse or opposite of an exponential function. This means that all exponential equations can be written in logarithmic form and vice versa. Where 𝑥 is greater than naught, 𝑎 is greater than naught, and 𝑎 is not equal to one, 𝑦 is equal to the log base 𝑎 of 𝑥 is equivalent to 𝑎 to the power of 𝑦 equals 𝑥. Any equation written in logarithmic form has an equivalent equation in exponential form. We will now look at a question where we need to change an equation in exponential form to one in logarithmic form.

Express four to the power of negative two equals one sixteenth in its equivalent logarithmic form.

Our initial equation is written in exponential form. This means that it is written 𝑎 to the power of 𝑥 is equal to 𝑦. We know that if 𝑎 to the power of 𝑥 is equal to 𝑦, then 𝑥 is equal to log base 𝑎 of 𝑦. Any exponential equation has an equivalent logarithmic equation. In this question, the value of 𝑎 is four, 𝑥 is equal to negative two, and 𝑦 is equal to one sixteenth. We can therefore conclude that negative two is equal to log base four of one sixteenth. The equivalent logarithmic form to the exponential equation four to the power of negative two equals one sixteenth is negative two is equal to log base four of one sixteenth.

In our next question, we’ll convert an equation in logarithmic form to its equivalent exponential form.

Express log base 20 of 𝑧 equals one-half in its equivalent exponential form.

We are given an equation in this question in logarithmic form. These can be written log base 𝑎 of 𝑦 is equal to 𝑥. We know that any equation in logarithmic form will have an equivalent equation in exponential form such that if log base 𝑎 of 𝑦 is equal to 𝑥, then 𝑦 is equal to 𝑎 to the power of 𝑥. In this question, the base 𝑎 is equal to 20, 𝑦 is equal to 𝑧, and 𝑥 is equal to one-half. The equation can therefore be rewritten as 𝑧 is equal to 20 to the power of a half. This is the equivalent exponential form to log base 20 of 𝑧 equals one-half.

Whilst it is not required in this question, we recall that 𝑥 to the power of a half is the same as the square root of 𝑥. This means that 20 to the power of a half is equal to the square root of 20 which, using our laws of radicals or surds, simplifies to two root five. 𝑧 is equal to two root five. As we were just asked to give our answer in exponential form though, the answer is 𝑧 is equal to 20 to the power of a half.

In our next two questions, we will deal with logarithmic form to the base 10. This is known as the common logarithm.

Express 10 cubed equals 1000 in its equivalent logarithmic form.

The equation that we are given in this question is written in exponential form of the type 𝑎 to the power of 𝑥 is equal to 𝑦. We know that any equation in exponential form has an equivalent logarithmic form. If 𝑎 to the power of 𝑥 is equal to 𝑦, then 𝑥 is equal to log base 𝑎 of 𝑦. In this question, the base 𝑎 is equal to 10, the exponent 𝑥 is equal to three, and 𝑦 is equal to 1000. This means that we can rewrite the equation as three is equal to log base 10 of 1000.

We recall that log base 10 is called the common logarithm. This means that when a logarithm is written without a base, it is assumed to be base 10. The log button that can be found on some scientific calculators is log to the base 10. When working in base 10, we do not need to write the base. Therefore, log of 1000 is equal to three. This is the equivalent logarithmic form to 10 cubed or 10 to the power of three is equal to 1000.

Express log of one million equals six in its equivalent exponential form.

The equation we are given is written in logarithmic form, which has a general form of log base 𝑎 of 𝑦 is equal to 𝑥. We know that any equation in logarithmic form has an equivalent equation in exponential form. If log base 𝑎 of 𝑦 is equal to 𝑥, then 𝑦 is equal to 𝑎 to the power of 𝑥. We notice in this question that there is no base. When any logarithm is shown without a base, it is assumed to be base 10. This is known as the common logarithm. We can therefore see that 𝑎 is equal to 10, 𝑦 is equal to one million, and 𝑥 is equal to six.

Rewriting the equation in exponential form, we have one million is equal to 10 to the sixth power or 10 to the power of six. We know this answer is correct as 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 is equal to one million. When raising 10 to a power, the exponent corresponds to the number of zeros.

In our final two questions, we will be dealing with the natural logarithm.

Write the exponential equation 𝑒 to the power of 𝑥 equals five in logarithmic form.

In order to answer this question, we need to recall the definition of the natural logarithm. When 𝑒 to the power of 𝑦 is equal to 𝑥, then base 𝑒 logarithm of 𝑥 is log base 𝑒 of 𝑥 which is written ln of 𝑥 which equals 𝑦. The natural logarithm function is the inverse of the exponential function. In this question, we are given the exponential equation 𝑒 to the power of 𝑥 is equal to five. Rewriting this in logarithmic form, we have 𝑥 is equal to the natural logarithm of five. The exponential equation 𝑒 to the power of 𝑥 equals five written in logarithmic form is 𝑥 is equal to ln of five.

Write the logarithmic equation eight equals ln of 𝑥 in exponential form.

In this question, we are given the natural logarithm ln of 𝑥. We know that the natural logarithm ln of 𝑥 is equal to log of base 𝑒 of 𝑥. We also know that ln is the inverse function of the exponential function. If 𝑦 is equal to ln of 𝑥, then 𝑒 to the power of 𝑦 equals 𝑥. In this question, eight is equal to ln of 𝑥. This means that 𝑒 to the power of eight or 𝑒 to the eighth power is equal to 𝑥. The logarithmic equation eight equals ln 𝑥 written in exponential form is 𝑒 to the power of eight equals 𝑥.

We will now summarize the key points from this video. We began this video by recalling that a logarithmic function is the inverse of an exponential function. This means that every logarithmic equation has an equivalent exponential equation. If 𝑥 is equal to log base 𝑎 of 𝑦, then 𝑎 to the power of 𝑥 is equal to 𝑦. This allows us to convert from a logarithmic equation to an exponential equation and vice versa. We also saw that a logarithm written without a base is assumed to be base 10. This is known as the standard logarithm.

We also saw that the natural logarithm is expressed ln of 𝑥. This is the same as log base 𝑒 of 𝑥. The natural logarithm function is the inverse of the exponential function. This means that if 𝑦 is equal to ln of 𝑥, then 𝑒 to the power of 𝑦 equals 𝑥.