Solving Logarithm Ratios to Find abc Value

The question asks us to find the value of \(abc\) given the relationship between logarithms:

\[ \frac{\log_{10}a}{b-c}=\frac{\log_{10}b}{c-a}=\frac{\log_{10}c}{a-b} \]

where \(a \ne b \ne c\).

This type of problem involves proportional relationships and properties of logarithms. We can solve this by setting the common ratio equal to a constant, say \(k\).

Let:

\[ \frac{\log_{10}a}{b-c}=\frac{\log_{10}b}{c-a}=\frac{\log_{10}c}{a-b} = k \]

From this equality, we can write three separate equations:

• Equation 1: \(\log_{10}a = k(b-c)\)
• Equation 2: \(\log_{10}b = k(c-a)\)
• Equation 3: \(\log_{10}c = k(a-b)\)

Now, let's consider the sum of the logarithms. Summing the left-hand sides of the three equations gives:

\[ \log_{10}a + \log_{10}b + \log_{10}c \]

Summing the right-hand sides gives:

\[ k(b-c) + k(c-a) + k(a-b) \]

So, we have:

\[ \log_{10}a + \log_{10}b + \log_{10}c = k(b-c + c-a + a-b) \]

Inside the parenthesis on the right side, the terms cancel out:

\[ b-c + c-a + a-b = (b-b) + (c-c) + (a-a) = 0+0+0 = 0 \]

Therefore, the equation becomes:

\[ \log_{10}a + \log_{10}b + \log_{10}c = k(0) = 0 \]

Now, we use the logarithm property that the sum of logarithms is the logarithm of the product: \(\log_B X + \log_B Y + \log_B Z = \log_B (XYZ)\). Applying this to the left side:

\[ \log_{10}(abc) = 0 \]

To find the value of \(abc\), we need to convert this logarithmic equation into an exponential equation. The definition of a logarithm states that if \(\log_B N = x\), then \(B^x = N\). In our case, the base \(B=10\), the exponent \(x=0\), and the number \(N=abc\).

So, converting \(\log_{10}(abc) = 0\) to exponential form gives:

\[ abc = 10^0 \]

Any non-zero number raised to the power of zero is equal to 1. Therefore,

\[ abc = 1 \]

The value of \(abc\) is 1.

Revision Table: Key Logarithm Properties Used

Additional Information on Logarithm Properties and Ratios

Understanding logarithm properties is crucial for solving equations involving logarithms. The property used here, the sum of logarithms, is a direct consequence of the exponent rules, as logarithms are essentially inverse operations to exponentiation.

When dealing with equal ratios like \(\frac{P}{Q} = \frac{R}{S} = \frac{T}{U}\), one common method is to set the common ratio equal to a constant \(k\). This allows you to express the numerators in terms of \(k\) and the denominators (\(P = kQ\), \(R = kS\), \(T = kU\)). Then, you can manipulate these new equations to find the desired value or relationship, often by adding, subtracting, or multiplying the equations strategically, as demonstrated in this problem by summing the logarithms.

The condition \(a \ne b \ne c\) is important because if any of the denominators \((b-c)\), \((c-a)\), or \((a-b)\) were zero, the original ratios would be undefined unless the corresponding numerator was also zero, which would lead to \(\log_{10}a = 0\), \(\log_{10}b = 0\), or \(\log_{10}c = 0\), meaning \(a=1\), \(b=1\), or \(c=1\). The condition \(a \ne b \ne c\) ensures the denominators are non-zero and the ratios are well-defined for distinct \(a, b, c\).