Expansión de logaritmos

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Lluís Enric Mayans
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Expansión de logaritmos

When you are asked to expand log expressions, your goal is to express a single logarithmic expression into many individual parts or components. This process is the exact opposite of condensing logarithms because you compress a bunch of log expressions into a simpler one.

The best way to illustrate this concept is to show a lot of examples. In this lesson, there are eight worked problems.

The key to successfully expanding logarithms is to carefully apply the rules of logarithms. Take time to go over the rules and understand what they are trying to “say”.

Por ejemplo, Regla 1 se llama Regla del producto. What it does is break the product of expressions as a sum of log expressions. See the rest of the descriptions below.

Reglas de logaritmos

Estudie la descripción de cada regla para obtener una comprensión intuitiva de la misma.

Descripciones de las reglas de logaritmos

Regla 1: Regla de producto

El logaritmo del producto de números es la suma de logaritmos de números individuales.

Regla 2: Regla del cociente

El logaritmo del cociente de números es la diferencia del logaritmo de números individuales.

Regla 3: Regla del poder

El logaritmo de un número exponencial es el exponente multiplicado por el logaritmo de la base.

Regla 4: Regla cero

El logaritmo de 1 con b> 0 pero b ne 1 es igual a cero.

Regla 5: Regla de identidad

The logarithm of a number that is equal to its base is just 1. Since the number is also the base, b, that means b>0 but b ne 1.

Regla 6: Regla del logaritmo del exponente

El logaritmo de un número exponencial donde su base es la misma que la base del logaritmo es igual al exponente.

Regla 7: exponente de la regla logarítmica

Raising the logarithm of a number by its base equals the number.

Examples of How to Expand Logarithms

ejemplo 1: Expand the log expression.

Looking inside the parenthesis, we see a product of a number and variables. The Product Rule doesn’t say that there should only be two factors inside, in fact, there could be more. Okay, so we’ll separate the main log expression as the sum of four logs.

ejemplo 2: Expand the log expression.

The inside of the parenthesis is a fraction that means I will first apply the Quotient Rule. Since the numerator is a product of 7 and x, I use the Product Rule to break it up.

ejemplo 3: Expand the log expression.

Okay, this one is also in fraction form so the Quotient Rule is the first step that we’re going to apply.

Now, there’s something “new” in this problem. That is, the numerator contains a variable with an exponent. This should be easy since Rule 3 or Power Rule can easily handle it. Just bring the exponent down to the left, that’s it!

In addition, there’s a radical expression in the denominator. Remember that a radical can be expressed as a fractional exponent. Since this radical is a square root that means the power is just large{1 over 2}.

Also, this is the first time we see Rule 5 or Identity Rule of Logarithm in action. Expect to apply this rule more often because it is extremely useful in the simplification process.

ejemplo 4: Expand the log expression.

Don’t let that square root symbol scare you. Just think of it as the power or exponent of large{1 over 2}. So this problem is reduced to expanding a log expression with a power of large{1 over 2}. This is where the Power Rule brings down that exponent large{1 over 2} to the left of the log, and then you expand the rest as usual.

ejemplo 5: Expand the log expression.

This problem is quite interesting because the entire expression is being raised to some power. In addition, the presence of a square root on the numerator adds some level of difficulty.

However, if we stick to the basics by carefully applying the rules of exponents in every step, we should have no problem conquering this problem! Let’s plan out our course of action, okay?

Bring down that exponent 4 using the Power Rule. Next, use the Quotient Rule to express the fraction as a difference of log expressions. And finally, don’t forget that square root is just a fractional exponent of large{1 over 2}.

I distributed the 4 into the grouping symbol to get rid of the fraction large{1 over 2}. It’s also okay to leave the 4 outside the grouping symbol which means we don’t have to distribute the 4 into the expressions inside the square brackets. Either of the two answers should be correct.

ejemplo 6: Expand the log expression.

It’s just a cube root of some rational expression. Replace the cube root symbol by a fractional power of large{1 over 3}. That exponent of large{1 over 3} can be brought down using the Power Rule of Logarithm.  Now we have just to deal with the rational expression using the Quotient Rule, then finish it off using the Product Rule.

Don’t get distracted by the different types of grouping symbols. The idea is to make sure that we are applying the logarithm rules correctly in each step that we undertake, without committing algebraic mistakes such as distributing -1 into the grouping symbol.

ejemplo 7: Expand the log expression.

I would immediately apply the Product Rule to separate the factors into a sum of logarithmic terms. Numerically simplify the {log _4}64 since 64 can be expressed as an exponential number with base 4, that is 64 = {4^3}. Deal with the square roots by replacing them with fractional power then use the Power Rule of logarithms to bring it down in front of the log symbol as a multiplier.

ejemplo 8: Expand the log expression.

The main power 3 can be placed out front as multiplier using the Power Rule. The Quotient rule should deal with the fractional expressions by writing them as the difference of logs. Now, replace the square root symbol with the exponent of large{1 over 2}. Similarly, replace the cube root symbol with the exponent of large{1 over 3}.

Practica con hojas de trabajo

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Reglas de logaritmos

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