Question
Write the equation in logarithmic form. Assume that all constants are positive and not equal to 1 . \( 10^{c}=u \) \( \square \) Hint
Ask by Vaughn Lee. in the United States
Jan 29,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The logarithmic form of \( 10^{c} = u \) is \( \log_{10} u = c \).
Solution
To convert the exponential equation \( 10^{c} = u \) into logarithmic form, you can use the definition of a logarithm. The logarithm base 10 of \( u \) is equal to the exponent \( c \) that you need to raise 10 to in order to get \( u \).
**Logarithmic Form:**
\[
\log_{10} u = c
\]
**Explanation:**
- **Base:** The base of the logarithm is 10, which corresponds to the base of the exponential equation.
- **Argument:** The argument of the logarithm is \( u \), which is the result of the exponential expression.
- **Result:** The logarithm equals \( c \), the exponent in the original exponential equation.
So, the equation \( 10^{c} = u \) written in logarithmic form is:
\[
\log_{10} u = c
\]
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Simplify this solution Bonus Knowledge
To convert the equation \( 10^{c}=u \) into logarithmic form, you would write it as \( \log_{10}(u) = c \). This tells you that \( c \) is the exponent to which the base 10 must be raised to produce \( u \). It’s quite a neat way to flip the equation on its head! Another fun fact is that logarithms can be incredibly useful in various fields, such as science and finance. For instance, in the field of sound, the decibel scale (a logarithmic scale) helps us measure sound intensity. Instead of using large numbers, we can express the vast range of sound in a much more manageable way!
