Question

Condense the logarithmic expression. $$2 \log x+\log 11$$

Short answer

The condensed form of the given expression is \(\log (11x^2)\)

Step-by-step solution

Identify shared base log

As all the logs in the exercise are to the base 10 and have positive arguments, the product and power laws of logarithms are applicable here. This is a very important first observation.

Apply logarithm power rule

The power rule of logarithms (which is \(\log_b(M^n)=n \log_b(M)\)) is applied in reverse to the term \(2 \log x\), i.e., it can be rewritten as \(\log (x^2)\), because the coefficient 2 becomes the exponent of x.

Apply logarithm product rule

The product rule of logarithms is then applied to the expression \(\log(x^2) + \log (11)\), which can be combined to give \(\log (11x^2)\), as the sum of the log of two terms can be combined to be a single log of their product.

Key concepts

Logarithm Power Rule

The logarithm power rule is a pivotal concept when working with logarithmic expressions. The rule itself is quite simple: for any positive number M, base b (where b is not equal to 1), and exponent n, the power rule states that \( \log_b(M^n)=n\cdot\log_b(M) \.\)

This property allows us to move between expressions with exponents inside a logarithm and those with coefficients in front of a logarithm, which is especially useful in solving logarithmic equations or simplifying expressions.

Consider how this applies to condensing logarithmic expressions. When we have an expression like \(2\log(x)\), we can interpret the coefficient '2' as an exponent of 'x' inside the logarithm, based on the power rule. Thus, \(2\log(x)\) is equivalent to \(\log(x^2)\). It 'condenses' the expression by removing the coefficient in front and encasing the power inside the logarithm, making the expression neater and often easier to work with.

Logarithm Product Rule

Expanding on the foundational knowledge of logarithms, another vital tool in our mathematical toolkit is the logarithm product rule. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors—if we have two positive numbers M and N, and a base b, the rule is formalized as \( \log_b(MN) = \log_b(M) + \log_b(N) \.\)

Applying this in practice transforms separate logarithms into a single expression, facilitating easier calculations and further simplifications. In the context of our exercise, we have \(\log(x^2) + \log(11)\), and utilizing the product rule, we merge these into one log expression \(\log(11x^2)\).

The product rule is particularly effective for condensing expressions where logarithms are added together. Recognizing when to apply this rule is a great skill that simplifies problems, especially when dealing with complex logarithmic equations.

Base 10 Logarithms

When we refer to 'base 10 logarithms', we are talking about logarithms that use 10 as their base; these are also known as 'common' logarithms. The notation for a base 10 logarithm is typically \(\log(x)\), omitting the base for simplicity, as it's understood to be 10.

In real-world scenarios and certain mathematical contexts—especially in scientific calculations—base 10 logarithms are extensively used because they relate to our decimal number system. For example, the pH scale in chemistry and the Richter scale for earthquake intensity both use base 10 logarithms.

Understanding base 10 logarithms is essential when solving exercises like the one provided, where we do not see any base indicated. This is because the assumption is that we're using common logarithms, and as per their properties, they follow the same product and power rules as logarithms with other bases. Hence, we can confidently proceed with simplification under this assumption, just as was done in the solution to achieve the final condensed expression \(\log(11x^2)\).