Calculate Logarithms: Log(28)51, Log(70)69, Log(85)46

by Jhon Lennon 54 views

Alright, guys, let's dive into the fascinating world of logarithms! We're going to break down how to calculate the values of log base 28 of 51, log base 70 of 69, and log base 85 of 46. Plus, we'll figure out what that '0' at the end might be hinting at. So, buckle up and get ready for some logarithmic fun!

Understanding Logarithms

Before we jump into the calculations, let's make sure we're all on the same page about what a logarithm actually is. In simple terms, a logarithm answers the question: "To what power must I raise this base to get this number?" For example, log base 10 of 100 (written as log₁₀(100)) is 2, because 10² = 100. The base is the small number written below the 'log' (like the 10 in log₁₀), and the number we're trying to reach is the argument (like the 100 in log₁₀(100)).

The general form of a logarithm is: logₐ(b) = x, which means aˣ = b. Here, 'a' is the base, 'b' is the argument, and 'x' is the logarithm (the power to which we raise 'a' to get 'b'). Understanding this relationship is crucial for tackling any logarithm problem. Logarithms are essentially the inverse of exponential functions, which makes them incredibly useful in various fields like science, engineering, and finance. They help us deal with very large or very small numbers in a more manageable way.

Also, it's super important to remember a few key properties of logarithms. For instance, logₐ(1) = 0 for any base 'a' because any number raised to the power of 0 is 1. Another handy property is logₐ(a) = 1 because any number raised to the power of 1 is itself. These properties can often simplify complex logarithmic expressions, making them easier to solve. So, keep these in your back pocket as we move forward!

Calculating log(28)51

Let's kick things off with calculating log base 28 of 51. Now, since most calculators don't have a direct way to calculate logs with arbitrary bases, we're going to use the change of base formula. This formula is a lifesaver and allows us to convert any logarithm into a form that our calculators can handle. The change of base formula is: logₐ(b) = logₓ(b) / logₓ(a), where 'x' can be any base, but we usually use base 10 (log) or base e (ln) because those are readily available on calculators.

So, let's use base 10 for this calculation. We can rewrite log₂₈(51) as log₁₀(51) / log₁₀(28). Now, grab your calculator and find the values of log₁₀(51) and log₁₀(28). You should get approximately:

log₁₀(51) ≈ 1.70757 log₁₀(28) ≈ 1.44716

Now, simply divide the first value by the second: 1.70757 / 1.44716 ≈ 1.1799. Therefore, log₂₈(51) ≈ 1.1799. What this means is that 28 raised to the power of approximately 1.1799 equals 51. You can check this by calculating 28^1.1799 on your calculator, and you should get a value very close to 51. Isn't that neat?

In summary, to find log base 28 of 51, we used the change of base formula to convert it into base 10 logarithms, which we could then easily calculate using a calculator. This method is super versatile and can be applied to any logarithm with any base. Remember to always double-check your calculations to ensure accuracy, and you'll be a logarithm pro in no time!

Calculating log(70)69

Next up, we're tackling log base 70 of 69. Just like before, we'll use the change of base formula to make this calculation manageable. The formula, as a quick reminder, is logₐ(b) = logₓ(b) / logₓ(a). We'll stick with base 10 logarithms for simplicity, but feel free to use natural logarithms (base e) if you prefer – the result will be the same!

So, we rewrite log₇₀(69) as log₁₀(69) / log₁₀(70). Now, let's find those values using a calculator:

log₁₀(69) ≈ 1.83885 log₁₀(70) ≈ 1.84510

Now, divide the first value by the second: 1.83885 / 1.84510 ≈ 0.99661. Therefore, log₇₀(69) ≈ 0.99661. This tells us that 70 raised to the power of approximately 0.99661 equals 69. Again, you can verify this by calculating 70^0.99661 on your calculator, and you should see a value very close to 69.

You might notice that this value is very close to 1. This makes sense because 69 is very close to 70. Since logₐ(a) = 1, when the argument is close to the base, the logarithm will be close to 1. This kind of intuition can be really helpful in quickly estimating logarithm values and checking if your calculated answers are reasonable.

In this case, log base 70 of 69 is approximately 0.99661. Remember, the change of base formula is your best friend when dealing with logarithms that don't have a common base. Keep practicing, and you'll become super comfortable with these calculations!

Calculating log(85)46

Alright, let's move on to our final logarithm: log base 85 of 46. As you've probably guessed, we're going to use the trusty change of base formula once again. This formula really is the key to unlocking logarithms with any base, so it's worth memorizing if you haven't already.

We'll rewrite log₈₅(46) as log₁₀(46) / log₁₀(85). Time to pull out the calculator and find those base 10 logarithm values:

log₁₀(46) ≈ 1.66276 log₁₀(85) ≈ 1.92942

Now, let's divide: 1. 66276 / 1.92942 ≈ 0.86180. So, log₈₅(46) ≈ 0.86180. This means that 85 raised to the power of approximately 0.86180 equals 46. As always, feel free to check this on your calculator by calculating 85^0.86180, and you should get a value close to 46.

In this case, the logarithm is less than 1, which makes sense because 46 is less than 85. When the argument is smaller than the base, the logarithm will be less than 1. This is another useful piece of intuition that can help you quickly assess whether your calculated logarithm values are in the right ballpark.

To recap, we found that log base 85 of 46 is approximately 0.86180. By now, you should be feeling pretty confident with using the change of base formula to calculate logarithms with different bases. Keep up the great work!

Interpreting the '0'

Now, let's address that '0' at the end of your original request. It's a bit unclear what the '0' is supposed to represent in this context without more information. However, here are a few possibilities:

  1. Rounding: The '0' might be indicating that you want the answers rounded to the nearest whole number. In that case:

    • log₂₈(51) ≈ 1
    • log₇₀(69) ≈ 1
    • log₈₅(46) ≈ 1

  2. Equation to Solve: It could be part of an equation that you want to solve. For example, if the full expression was log₂₈(51) + log₇₀(69) + log₈₅(46) = x, then we would sum our calculated values to find 'x'. But since we only have a '0', it's hard to say.

  3. Contextual Meaning: It might have a specific meaning within a larger problem or application. Without knowing the context, it's difficult to give a precise interpretation.

If you can provide more information about the context of the '0', I'd be happy to give you a more accurate explanation. In the meantime, I hope this comprehensive breakdown of calculating the logarithms has been helpful! Remember, practice makes perfect, so keep working with logarithms, and you'll become a true master of these mathematical tools.