Question

Simplify each expression. $$10^{\log x^{2}}$$

Short answer

\(10^{\log x^{2}} = x^{2}\).

Step-by-step solution

Understand the Properties of Logarithms

Recall that one of the fundamental properties of logarithms is that for any positive numbers a and b, and any positive real number x, the expression \(a^{\log_{a} (b)}\) is equal to b. Here, we are working with the base 10 logarithm.

Apply the Logarithmic Property to Simplify

Using the aforementioned property, take the base of the logarithm (which is 10) and raise it to the power of the logarithm itself, \(10^{\log_{10} (x^2)}\). This simplifies directly to \(x^2\) because the base and the logarithm cancel each other out.

Key concepts

Simplifying Logarithmic Expressions

Dealing with logarithmic expressions might seem daunting at first, but understanding a few key principles can make simplification a breeze. Essentially, simplifying logarithms involves rewriting them in a more straightforward or easily interpretable form without altering their value. What makes this possible is the inherent relationship between logarithms and exponents.

For example, in the expression \(10^{\log x^{2}}\), simplification is achieved by knowing that the logarithmic function \(\log_{a}(b)\) is essentially asking for the power that \(a\) must be raised to, to yield \(b\). Hence, \(a^{\log_{a}(b)} = b\). This is why the given expression simplifies to \(x^{2}\), an exposition of the logarithm's ability to 'undo' the exponentiation when the bases match. Understanding this property is pivotal to mastering logarithmic expressions.

Properties of Logarithms

The properties of logarithms are mathematical tools that make working with logarithmic functions much more accessible. They can convert complex logarithmic equations into more manageable forms. The properties are grounded in the nature of logarithms as the inverses of exponential functions.

Key properties include the Product Rule, \(\log_{a}(mn) = \log_{a}(m) + \log_{a}(n)\), Quotient Rule, \(\log_{a}(\frac{m}{n}) = \log_{a}(m) - \log_{a}(n)\), and Power Rule, \(\log_{a}(m^{n}) = n\log_{a}(m)\). Additionally, understanding that \(\log_{a}(a) = 1\) and \(\log_{a}(1) = 0\) is crucial. These properties not only assist in simplification but also play a role in expanding logarithmic expressions when necessary.

Exponential and Logarithmic Equations

Exponential and logarithmic equations are two sides of the same coin, connected through their inverse relationship. To solve these equations, we often use logarithms to isolate the variable of interest, especially within an exponent.

For instance, if given \(a^{x} = b\), taking the logarithm of both sides, preferably with base \(a\), yields \(x = \log_{a}(b)\). Similarly, if you start with a logarithm, such as \(\log_{a}(x) = c\), you would 'exponentiate' both sides to solve for \(x\), resulting in \(x = a^{c}\). This back-and-forth between the logarithmic and exponential form allows for the solution of equations which might otherwise be insoluble using elementary algebraic techniques.

Mathematical Simplification

Mathematical simplification is akin to tidying up a room to make space and improve clarity. In the context of logarithms (or any mathematical concept), this means rewriting expressions in their least complex form, while preserving their integrity and solving equations efficiently.

Strategies for simplification include combining like terms, factoring, canceling common factors, and using identities (like those of logarithmic properties) to condense the expression. Simplification is not just a means to an end; it’s pivotal in understanding the underlying structure of mathematical statements, enhancing problem-solving abilities, and in the context of logarithms, recognizing patterns and relationships that might not be immediately obvious in their original form.