Complete the following table by rewriting the given exponential expressions to logarithmic form whenever possible and/or vice versa. Logarithmic Form 1. 3. 4. 5. Exponential Form 16- 24 9= √81 1=3-² 9 loga 9 = 2 logs 25
The Correct Answer and Explanation is :
Let’s start by converting the exponential expressions into logarithmic form and vice versa, one at a time, based on the given instructions.
1. ( 16 = 24 )
This equation is not written in exponential or logarithmic form clearly, as there is no base or exponent in this case. If we interpret the equation, we cannot directly convert it to logarithmic form. This equation appears to be a simple comparison (which is not fully clear from the way it’s written). Further clarification is needed.
2. ( \sqrt{81} )
We know that the square root of a number can be expressed as raising the number to the power of ( \frac{1}{2} ). So,
- Exponential form: ( 81^{\frac{1}{2}} )
Now, let’s convert this to logarithmic form. The logarithmic form of an exponential equation ( a^b = c ) is:
- Logarithmic form: ( \log_{81} \sqrt{81} = \frac{1}{2} )
3. ( 1 = 3^{-2} )
This is an exponential equation that we can rewrite as a logarithmic equation.
- Logarithmic form: ( \log_3 1 = -2 )
Since ( 3^{-2} = \frac{1}{9} ), and ( \log_3 \frac{1}{9} = -2 ), this is how the logarithmic form is obtained.
4. ( 9 )
This is written as a simple number, but without additional context or an equation, we can’t convert it to an exponential or logarithmic form.
5. ( \log_a 9 = 2 )
This is already in logarithmic form. To convert it to exponential form, we use the basic rule: if ( \log_b x = y ), then ( b^y = x ).
- Exponential form: ( a^2 = 9 )
6. ( \log_5 25 )
This is also in logarithmic form. The logarithmic expression ( \log_5 25 = 2 ) means that ( 5^2 = 25 ), so the exponential form is:
- Exponential form: ( 5^2 = 25 )
Summary of Conversion Rules:
- Exponential to Logarithmic form: ( a^b = c ) becomes ( \log_a c = b ).
- Logarithmic to Exponential form: ( \log_a c = b ) becomes ( a^b = c ).
This process is fundamental in understanding logarithmic and exponential relationships and is essential for solving equations in algebra and calculus, particularly when dealing with growth, decay, or any other phenomena that follow exponential patterns. Logarithms allow us to solve for unknown exponents, while exponentials help model real-world growth patterns.