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Chapter 5: Problem 35

Evaluate each expression. Do not use a calculator. $$\log 10^{1.5}$$

Short Answer

Expert verified
\(1.5\)

Step by step solution

01

Understanding the Expression

We need to evaluate the expression \( \log 10^{1.5} \). This expression involves a logarithm with base 10 (a common logarithm) and a power of 10 raised to 1.5.
02

Applying the Logarithmic Identity

Recall the identity for logarithms: \( \log_{b}(b^x) = x \). Here, our base \( b \) is 10 and \( x \) is 1.5. This identity tells us that \( \log 10^{1.5} = 1.5 \) because the base and the base of the exponent match.
03

Conclusion

Since \( \log 10^{1.5} = 1.5 \), we have evaluated the expression successfully using the property of logarithms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Identity
Logarithmic identities are useful tools in simplifying and solving expressions involving logarithms. The particular identity used in this exercise is \( \log_{b}(b^x) = x \). This means if you log a number that is a power of the base, you just get the exponent back.
This identity is valid because of the fundamental properties of exponents and logarithms. Since a logarithm is essentially the inverse of exponentiation, it 'undoes' what the exponent did.
  • If you have \( b^x \), then taking the logarithm with base \( b \) gives you back \( x \).
  • This makes it very handy for arithmetic involving powers of ten, such as in scientific notation.
In the exercise given, since we have a logarithm with the base of ten (\( \log 10^{1.5} \)), the base matches the base of the exponent. Therefore, using the identity, we know the result immediately is 1.5.
Common Logarithm
Common logarithms are simply logarithms with base 10. You will often see them written as \( \log \) without a base mentioned. This is a widely used base in many practical applications.
  • Think of common logarithms as designed for use with our decimal system, which also operates on base 10.
  • They are commonly used in science and engineering, especially when dealing with orders of magnitude.
For example, the logarithmic scale used in measuring sound intensity, known as decibels, relies on common logarithms.
The exercise involves a common logarithm, \( \log 10^{1.5} \). Here we know that because the base is 10, we can directly use the properties of common logarithms to simplify this expression, arriving comfortably at the answer of 1.5.
Exponentiation
Exponentiation is the operation of raising one number, known as the base, to the power of another number, known as the exponent. The notation \( b^x \) means multiplying the base \( b\) by itself \( x \) times.
Understanding exponentiation helps in assessing expressions like \( 10^{1.5} \).
  • Here, an exponent of 1.5 involves raising 10 to the power of one, and then further multiplying by the square root of 10 (as the .5 or half exponent signifies a root).
  • Essentially, exponentiation with non-integer exponents involves roots and powers, rather than just repeated multiplication.
In the context of logarithms, exponentiation is what a logarithm resolves. For instance, when we see \( \log 10^{1.5} \), we recognize this "undoes" the exponentiation according to the logarithmic identity, capturing back the exponent 1.5 directly and simply.

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