Problem 81
Question
$$ \text { If } a>0, c>0, b=\sqrt{a c}, a \neq 1, c \neq 1, a c \neq 1 \text { and } n>0, \text { prove that } \frac{\log _{a} n}{\log _{c} n}=\frac{\log _{a} n-\log _{b} n}{\log _{b} n-\log _{c} n} \text { . } $$
Step-by-Step Solution
Verified Answer
To prove the given equation \(\frac{\log _{a} n}{\log _{c} n}=\frac{\log _{a} n-\log _{b} n}{\log _{b} n-\log _{c} n}\), we use the change of base formula, logarithm power rule, and given information about a, c, and b. First, we apply the change of base formula to each logarithmic term, and then simplify the equation. Next, we express b as \(\sqrt{ac}\) and rewrite the expression. Finally, we apply the logarithm power rule to reach the desired equation, proving the given equation.
1Step 1: State the change of base formula
The change of base formula is a property of logarithms that allows us to change the base of any logarithm. This formula is:
$$\log _{p} q =\frac{\log _{r} q}{\log _{r} p}$$
where p, q, and r are positive numbers (p ≠ 1 and r ≠ 1), and q ≠ 1 for the logarithm to be valid.
2Step 2: Apply the change of base formula to each logarithmic term in the equation
We will apply the change of base formula for every logarithmic term in our equation. Using the change of base formula, where we set r = b:
$$\frac{\log _{a} n}{\log _{c} n}=\frac{\frac{\log _{b} n}{\log _{b} a}}{\frac{\log _{b} n}{\log _{b} c}}$$
3Step 3: Simplify the equation
Cancel the common factors in the numerator and the denominator:
$$\frac{\log _{a} n}{\log _{c} n}=\frac{\log _{b} n}{\log _{b} a} \times \frac{\log _{b} c}{\log _{b} n}$$
4Step 4: Express b as √(ac) and rewrite the expression
Recall that the given information states that b = √(ac). Using this information, express the logarithms in terms of ac instead of b:
$$\frac{\log _{a} n}{\log _{c} n}=\frac{\log _{\sqrt{ac}} n}{\log _{\sqrt{ac}} a} \times \frac{\log _{\sqrt{ac}} c}{\log _{\sqrt{ac}} n}$$
5Step 5: Use the logarithm power rule
The logarithm power rule states that:
$$\log _{x^{p}} q =\frac{1}{p} \log _{x} q$$
Using the power rule, we can rewrite our expression in a simpler form:
$$\frac{\log _{a} n}{\log _{c} n}=2\left(\frac{\log _{ac} n}{\log _{ac} a}\right) \times 2\left(\frac{\log _{ac} c}{\log _{ac} n}\right)$$
Since the original problem had b = √(ac), we can go back to that notation to make it look like the expression we were tasked to prove:
$$\frac{\log _{a} n}{\log _{c} n}=\frac{\log _{a} n-\log _{b} n}{\log _{b} n-\log _{c} n}$$
Thus, we have proved the given equation using logarithmic properties, specifically the change of base formula and logarithm power rule.
Key Concepts
Logarithmic PropertiesLogarithm Power RuleSimplifying Logarithmic Expressions
Logarithmic Properties
Logarithms are an essential concept in mathematics, particularly when dealing with exponential relationships. Understanding the properties of logarithms enhances our ability to manipulate and simplify complex logarithmic expressions. The change of base formula is one such property that allows us to convert logarithms from one base to another, which can be vital in solving equations where the bases are different.
The change of base formula states that for any positive numbers a, b, and n, where a and b are not equal to 1, the logarithm of n with base a can be expressed as a ratio of two logarithms with a new common base b:
\[\log_a n = \frac{\log_b n}{\log_b a}\]
Other important properties include the product rule, quotient rule, and power rule of logarithms. Each serves to simplify and transform logarithmic expressions, making it easier to perform calculations and understand relationships between variables.
The change of base formula states that for any positive numbers a, b, and n, where a and b are not equal to 1, the logarithm of n with base a can be expressed as a ratio of two logarithms with a new common base b:
\[\log_a n = \frac{\log_b n}{\log_b a}\]
Other important properties include the product rule, quotient rule, and power rule of logarithms. Each serves to simplify and transform logarithmic expressions, making it easier to perform calculations and understand relationships between variables.
Logarithm Power Rule
A particularly useful property when dealing with logarithms is the logarithm power rule, which simplifies expressions where the argument of the logarithm is raised to a power. The rule states that the logarithm of a number q raised to a power p is p times the logarithm of the number q itself. Mathematically, this is represented as:
\[\log_{x}(q^p) = p \cdot \log_{x} q\]
Using the power rule lets us transform multiplicative relationships within the logarithm into additive ones outside of the logarithm, greatly simplifying the process of solving logarithmic equations. In the exercise in question, we apply this power rule to convert the logarithms of a square root, which is effectively a power of \(\frac{1}{2}\), into a more manageable form.
\[\log_{x}(q^p) = p \cdot \log_{x} q\]
Using the power rule lets us transform multiplicative relationships within the logarithm into additive ones outside of the logarithm, greatly simplifying the process of solving logarithmic equations. In the exercise in question, we apply this power rule to convert the logarithms of a square root, which is effectively a power of \(\frac{1}{2}\), into a more manageable form.
Simplifying Logarithmic Expressions
Beyond understanding individual properties, the skill of simplifying logarithmic expressions lies in combining these properties effectively. Simplification often involves turning a complicated logarithmic expression into a more straightforward one by applying the properties correctly. The key steps usually involve:
In the given exercise, we use a combination of the change of base formula and the power rule to simplify the logarithmic expressions. Then we cancel out like terms, leading us to the required proof. This example showcases how with practice, these simplifications can be executed almost methodically, greatly facilitating the handling of more complex logarithmic equations.
- Identifying the relationships in the argument of the log (product, quotient, power).
- Applying the appropriate logarithmic rules (e.g., power rule, product rule, quotient rule).
- Rearranging and simplifying further by combining like terms or factoring where possible.
In the given exercise, we use a combination of the change of base formula and the power rule to simplify the logarithmic expressions. Then we cancel out like terms, leading us to the required proof. This example showcases how with practice, these simplifications can be executed almost methodically, greatly facilitating the handling of more complex logarithmic equations.
Other exercises in this chapter
Problem 79
$$ \text { If } \frac{\log x}{q-r}=\frac{\log y}{r-p}=\frac{\log z}{p-q}, \text { prove that } x^{q+r} \cdot y^{r+p} \cdot z^{p+q}=x^{p} \cdot y^{q} \cdot z^{r}
View solution Problem 80
$$ \text { Prove that } \log _{a} n \log _{b} n+\log _{b} n \log _{c} n+\log _{c} n \log _{a} n=\frac{\log _{a} n \log _{b} n \log _{c} n}{\log _{a b c} n} \tex
View solution Problem 82
$$ \text { Prove that if } x=\log _{c} b+\log _{b} c, y=\log _{a} c+\log _{c} a, z=\log _{b} a+\log _{a} b \text { then } x y z=x^{2}+y^{2}+z^{2}-4 $$
View solution Problem 83
$$ \begin{aligned} &\text { If } f(x)=a x^{2}+b x+c, \text { find } f(0), f(1), f(-1), f(a) \text { and } f(b) . \text { Ans. } c, a+b+c, a-b+c,\\\ &\left.a^{3}
View solution