Express as a single logarithm

Question

Create a seamless, crisp and explanatory image that illustrates the mathematical concept of converting multiple logarithms into a single logarithm. Show two separate logarithmic expressions indicative of 2log3 and 4log2 on the left side of the image. Next, indicate a transformation process illustrated by a horizontal line leading to the right side of the image, but do not provide the final answer or any text. The illustration style must be clear, precise, and educational in nature. Use bright, soothing colors to make the content more attractive.

Express as a single logarithm

2log3 + 4log2

I'm confused as to how to do this, I keep getting the wrong answer. Thank you!

Answer 1

a log(b) = log(b^a)

log(a) + log(b) = log(a*b)

log(3^2) + log(2^4) = log(9 * 16)

Answer 2

Well, let me "log" into this problem and help you out! To express the expression 2log3 + 4log2 as a single logarithm, we can use the logarithm properties.

First, let's rewrite the expression using the product rule of logarithms:

2log3 + 4log2 = log3^2 + log2^4

Then, applying the power rule of logarithms:

= log9 + log16

Finally, we can combine the logarithms using the sum rule:

= log9 * 16

Therefore, the expression 2log3 + 4log2 can be expressed as a single logarithm: log9 * 16.

Answer 3

To express 2log3 + 4log2 as a single logarithm, we can use the properties of logarithms. Specifically, we can use the following two rules:

1. The product rule: loga(b) + loga(c) = loga(b * c)
2. The power rule: n * loga(b) = loga(b^n)

Let's use these rules to simplify the expression:

2log3 + 4log2

Using the power rule, we can rewrite the expression as:

log3(2^2) + log2(4^4)

Simplifying the exponents, we get:

log3(4) + log2(256)

Now, using the product rule, we can combine the logarithms:

log3(4 * 256)

Simplifying the multiplication, we have:

log3(1024)

Therefore, 2log3 + 4log2 simplifies to log3(1024).

Answer 4

To express the expression 2log3 + 4log2 as a single logarithm, you can use the properties of logarithms. The two properties we will use are:

1. The product rule: logₐ(x * y) = logₐ(x) + logₐ(y).
2. The power rule: logₐ(x^y) = y * logₐ(x).

Let's apply these properties to the expression 2log3 + 4log2:

2log3 + 4log2

Using the product rule on both terms:

= log₃(3^2) + log₂(2^4)

Simplifying further:

= log₃(9) + log₂(16)

Now, to combine the two logarithms into a single logarithm, we can use the product rule again:

= log₃(9 * 2^4)

Simplifying:

= log₃(9 * 16)

Final Answer:

= log₃(144)

Therefore, 2log3 + 4log2 can be expressed as a single logarithm: log₃(144).

Remember to always apply logarithmic properties and simplify as much as possible to get the correct answer.