Calculating Log (7200) Given Log 7 = 0.845 A Step By Step Solution

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In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex calculations and unveiling hidden relationships between numbers. This article delves into the fascinating world of logarithms, specifically focusing on how to determine the value of log (7200) when provided with the information that log 7 equals 0.845. We will embark on a step-by-step journey, unraveling the properties of logarithms and applying them to solve this intriguing problem. Join us as we explore the intricacies of logarithms and discover how they can be used to decipher numerical mysteries.

At its core, a logarithm is the inverse operation of exponentiation. In simpler terms, if we have an equation like b^x = y, the logarithm of y to the base b is x. This can be written as log_b(y) = x. The base, b, is the foundation upon which the logarithm is built. In common logarithms, the base is 10, and we often omit writing the base explicitly, so log(y) is understood to mean log_10(y). Understanding this fundamental relationship between exponentiation and logarithms is crucial for navigating the world of logarithmic calculations.

Logarithms play a vital role in various fields, including science, engineering, and finance. They are particularly useful for handling very large or very small numbers, as they can compress these numbers into a more manageable scale. For instance, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. This allows us to represent a wide range of earthquake intensities using a relatively small set of numbers. Similarly, logarithms are used in chemistry to express the pH of solutions, which can vary over several orders of magnitude. In finance, logarithms are used to calculate compound interest and to analyze investment growth.

Key Properties of Logarithms: Our Guiding Principles

To effectively tackle logarithmic problems, it's essential to grasp the key properties that govern their behavior. These properties act as the rules of the logarithmic game, guiding us through manipulations and simplifications. Let's explore some of the most fundamental properties:

  1. Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as log_b(xy) = log_b(x) + log_b(y). This rule allows us to break down complex logarithmic expressions into simpler components, making calculations more manageable. For example, if we want to find the logarithm of 100, which can be written as 10 * 10, we can use the product rule to say log(100) = log(10 * 10) = log(10) + log(10) = 1 + 1 = 2.

  2. Quotient Rule: The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. This can be written as log_b(x/y) = log_b(x) - log_b(y). This rule is particularly helpful when dealing with fractions or division within logarithmic expressions. For instance, if we want to find the logarithm of 1/2, we can use the quotient rule to say log(1/2) = log(1) - log(2) = 0 - log(2). Since log(2) is approximately 0.301, log(1/2) is approximately -0.301.

  3. Power Rule: The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This is expressed as log_b(x^p) = p * log_b(x). This rule is incredibly useful for simplifying expressions involving exponents within logarithms. For example, if we want to find the logarithm of 2^3, we can use the power rule to say log(2^3) = 3 * log(2) = 3 * 0.301 = 0.903.

  4. Change of Base Rule: This rule allows us to convert logarithms from one base to another. It is expressed as log_a(x) = log_b(x) / log_b(a). The change of base rule is invaluable when we need to work with logarithms in a base that is not directly available on a calculator or in a table. For instance, if we want to find log_2(10), but we only have common logarithms (base 10) available, we can use the change of base rule to say log_2(10) = log_10(10) / log_10(2) = 1 / 0.301 = 3.322.

Understanding and applying these properties is the key to unlocking the power of logarithms and solving a wide range of mathematical problems. They provide a framework for manipulating logarithmic expressions, breaking them down into simpler components, and ultimately arriving at a solution. In the context of our problem, finding log(7200) given log(7) = 0.845, these properties will be our guiding principles.

Our challenge is to determine the value of log (7200), given that log 7 = 0.845. This problem exemplifies how we can leverage the properties of logarithms to solve seemingly complex calculations by breaking them down into simpler, manageable steps. The key lies in recognizing how we can express 7200 in terms of its prime factors, particularly 7, which is the only logarithmic value we have been provided.

To find log (7200), we need to express 7200 as a product of its prime factors. This will allow us to utilize the properties of logarithms, specifically the product rule, to break down the problem into smaller, more manageable parts. Let's begin by finding the prime factorization of 7200:

7200 = 72 * 100

Now, we can further break down 72 and 100 into their prime factors:

72 = 8 * 9 = 2^3 * 3^2 100 = 10 * 10 = 2^2 * 5^2

Combining these, we get the prime factorization of 7200:

7200 = 2^3 * 3^2 * 2^2 * 5^2 = 2^5 * 3^2 * 5^2

Unfortunately, we cannot directly express 7200 in terms of 7 using prime factorization. So, let's rewrite 7200 in a different way that might involve 7:

7200 = 7 * (7200/7) ≈ 7 * 1028.57

This approach doesn't seem to simplify our calculation effectively since 1028.57 is not easily factorizable into simple terms. Let's try a different approach. We can express 7200 as:

7200 = 7.2 * 1000

We know that log 1000 = log (10^3) = 3, but we still need to deal with 7.2. Let's express 7.2 as a fraction:

  1. 2 = 72 / 10

So, we can rewrite 7200 as:

7200 = (72/10) * 1000 = 72 * 100

We already found the prime factorization of 7200 as 2^5 * 3^2 * 5^2. Now we can apply logarithms to this expression:

log (7200) = log (2^5 * 3^2 * 5^2)

Using the product rule of logarithms, we can break this down into:

log (7200) = log (2^5) + log (3^2) + log (5^2)

Now, we apply the power rule of logarithms:

log (7200) = 5 * log (2) + 2 * log (3) + 2 * log (5)

We know that log 7 = 0.845, but we need log 2, log 3, and log 5. We can use the fact that:

log 10 = log (2 * 5) = log 2 + log 5 = 1

This means:

log 5 = 1 - log 2

Now we have:

log (7200) = 5 * log (2) + 2 * log (3) + 2 * (1 - log 2) log (7200) = 5 * log (2) + 2 * log (3) + 2 - 2 * log (2) log (7200) = 3 * log (2) + 2 * log (3) + 2

Unfortunately, we still need the values of log 2 and log 3. We cannot directly derive these values from the given information (log 7 = 0.845). However, we can use the approximate values of log 2 ≈ 0.3010 and log 3 ≈ 0.4771.

log (7200) ≈ 3 * 0.3010 + 2 * 0.4771 + 2 log (7200) ≈ 0.9030 + 0.9542 + 2 log (7200) ≈ 3.8572

Therefore, the approximate value of log (7200) is 3.8572.

Let's try a different approach to see if we can utilize the given log 7 = 0.845 more directly. We have 7200 = 72 * 100. We can rewrite 72 as 8 * 9 = 2^3 * 3^2. So,

7200 = 2^3 * 3^2 * 10^2

Applying logarithms:

log (7200) = log (2^3 * 3^2 * 10^2) log (7200) = log (2^3) + log (3^2) + log (10^2) log (7200) = 3 * log (2) + 2 * log (3) + 2 * log (10) log (7200) = 3 * log (2) + 2 * log (3) + 2

Again, we face the same issue of needing log 2 and log 3. We cannot directly obtain these values from log 7. However, we can approximate log 2 and log 3 and proceed as before.

log (7200) ≈ 3 * 0.3010 + 2 * 0.4771 + 2 log (7200) ≈ 0.9030 + 0.9542 + 2 log (7200) ≈ 3.8572

This alternative approach yields the same result, reinforcing our previous calculation.

In conclusion, we have successfully determined the value of log (7200), given that log 7 = 0.845. The key to solving this problem lies in understanding and applying the properties of logarithms, particularly the product and power rules. By breaking down 7200 into its prime factors and utilizing these rules, we were able to simplify the problem and arrive at a solution. While we couldn't directly use the given log 7 value, we demonstrated how to approach such problems by using approximate values for log 2 and log 3. This exploration highlights the power of logarithms as a tool for simplifying complex calculations and revealing the hidden relationships within numbers. The result we found for log (7200) is approximately 3.8572. This showcases the practical application of logarithmic principles in mathematical problem-solving.