Question

Express as a single logarithm with a coefficient of \(1 .\) Assume that the logarithms in each problem have the same base. $$\log x+\log y+\log z$$

Short answer

\(\log x + \log y + \log z = \log(xyz)\)

Step-by-step solution

Understanding the Logarithmic Property

To combine logarithms into a single logarithm, you can use the property that states \(\log(a) + \log(b) = \log(ab)\). This property allows you to multiply the arguments of the logarithms when they are being added together.

Applying the Property to the Given Expression

Given the expression \(\log x + \log y + \log z\), apply the property \(\log(a) + \log(b) = \log(ab)\) to combine the first two logarithms: \(\log x + \log y = \log(xy)\).

Combining All Logarithms

Now combine the result of the previous step with the remaining logarithm using the same property: \(\log(xy) + \log z = \log((xy)z)\).

Simplifying the Expression

Simplify the expression by multiplying the arguments inside the logarithm to get a single logarithm: \(\log((xy)z) = \log(xyz)\).

Key concepts

Logarithmic Properties

The properties of logarithms are the backbone for understanding how to work with logarithmic expressions. One fundamental property is the product rule, which states that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying their arguments, expressed as \(\log(a) + \log(b) = \log(ab)\).

Another key property is the quotient rule, which relates the difference of logarithms to the division of their arguments: \(\log(a) - \log(b) = \log(\frac{a}{b})\). Lastly, the power rule implies that a logarithm of an exponentiated argument can be simplified by bringing the exponent down as a coefficient: \(\log(a^n) = n \log(a)\). Learning to apply these properties not only helps in simplifying logarithms but also in solving complex logarithmic equations.

Logarithmic Expressions

Logarithmic expressions can seem complicated, but understanding their basic structure is helpful for simplifying them. A logarithmic expression consists of a base, usually assumed to be 10 when not explicitly stated, and an argument, which is the value you are taking the logarithm of.

The expression \(\log x\) signifies the power to which the base must be raised to obtain the value \(x\). When combining logarithms, it is critical that they all have the same base; otherwise, they are not directly combinable using the logarithmic properties. Whether you're adding, subtracting, or dealing with more complex operations within logarithms, keeping the base consistent ensures you can apply the rules correctly and reach a simplified form.

Simplifying Logarithms

Simplifying logarithms into a single expression can de-clutter complex equations and make them easier to solve. The step-by-step solution provided for the exercise \(\log x + \log y + \log z\) is a perfect example of simplifying by using the properties of logarithms.

You begin by grouping the logarithms two at a time, applying the product rule to combine them. Continue this process until you are left with a single logarithm. The equation simplifies to \(\log(xyz)\), with a coefficient of 1, meaning the expression represents the exponent needed to raise the base to get the product of \(x\), \(y\), and \(z\). This simplification is powerful because it can transform a bulky equation into a more manageable form for further mathematical processing or evaluation.