Question

Convert the following expressions to the indicated base. \(a^{1 / \log a}\) using base \(10,\) for \(a>0\) and \(a \neq 1\)

Short answer

Question: Convert the given expression to base 10: \(a^{\frac{1}{\log a}}\) Answer: \(a^{\frac{\log_{10} a}{\log_{10} (\log a)}}\) for \(a>0\) and \(a \neq 1\)

Step-by-step solution

Identify the given expression

The expression we have is \(a^{\frac{1}{\log a}}\) with base \(10\). The current base for the logarithm is not specified, but we'll need to change it to base 10.

Change of base formula for logarithms

To change the base of the logarithm, we can use the change of base formula: \(\log_{b}x = \frac{\log_{d}x}{\log_{d}b}\), where \(d\) is the new base.

Apply the change of base formula to the given expression

Using the change of base formula, we get: \(a^{\frac{1}{\log a}} = a^{\frac{\log_{10} a}{\log_{10} (\log a)}}\), where \(a>0\) and \(a \neq 1\)

Final answer

The final converted expression to base 10 is: \(a^{\frac{\log_{10} a}{\log_{10} (\log a)}}\) for \(a>0\) and \(a \neq 1\)

Key concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see a number like \(a^n\), the base \(a\) is multiplied by itself \(n\) times. This concept is crucial in mathematics, as it helps simplify expressions and solve complex problems. For example, \(2^3\) means \(2 \times 2 \times 2\), which equals 8. Here's how exponents work:

• the base is the number being multiplied,
• the exponent indicates how many times the base is used as a factor.

Exponents can also appear as fractions, where they represent roots or powers. For example, \(a^{1/2}\) is the square root of \(a\). Understanding exponents is essential for solving problems involving growth, decay, and other mathematical operations.

Logarithms

Logarithms are the inverse operation of exponents. They help us find the power or exponent needed to obtain a certain number. When you see a logarithm \(\log_b(x)\), it asks: "What power do we need to raise the base \(b\) to in order to get \(x\)"? For instance, \(\log_2(8)\) equals 3, because \(2^3 = 8\).Key points about logarithms:

• They can convert multiplication into addition, which simplifies calculations.
• The base of a logarithm (like base 10) is significant because it indicates the scale of calculations.

Logarithms are very useful in many scientific fields, including biology, chemistry, and physics. They can help us understand phenomena that grow exponentially, such as population growth or radioactive decay.

Base 10

Base 10, also known as the decimal system, is the number system most commonly used by humans. It is based on the number 10, due to counting with ten fingers. In Base 10, each digit in a number has a place value that is a power of 10. For example, in the number 452, the digit 4 is in the 'hundreds' place, or \(4 \times 10^2\).The significance of Base 10:

• This base is intuitive and widely used in everyday life, including money and measurements.
• Many logarithmic calculations often use base 10, as in the case of common logarithms \(\log_{10}(x)\).

Knowing how to manipulate numbers in Base 10 is crucial for understanding a variety of mathematical concepts, especially when dealing with logarithms and exponents in scientific contexts.