1) use the properties of logarithms to simplify the logarithmic expression.

log base 10 (9/300)

log - log 300
log 9 = 2 log 3
log 300 = log 3 + log 100 = log 3+2

I just do not know how to put these together now!

log base 10 (9/300)

= log10 (3/100)
= log10(3) - log10(100)
= log10(3) - 2

Further simplification is not possible.

The numerical value is 0.47712 - 2
= -1.52288

log a b * log bc * logc a=1

To simplify the logarithmic expression log base 10 (9/300), you can use the properties of logarithms.

First, recall the property log(a/b) = log(a) - log(b). Using this property, we can rewrite the expression as:

log base 10 (9/300) = log base 10 (9) - log base 10 (300)

Next, use another property of logarithms, log(a^b) = b*log(a), to simplify each individual logarithm:

log base 10 (9) - log base 10 (300) = 2*log base 10 (3) - (log base 10 (3) + log base 10 (100))

Finally, simplify further:

2*log base 10 (3) - (log base 10 (3) + log base 10 (100)) = log base 10 (3^2) - (log base 10 (3) + log base 10 (10^2))

= log base 10 (9) - (log base 10 (3) + 2)

Now, the expression is simplified using the properties of logarithms.

To simplify the logarithmic expression log base 10 (9/300), you can use the properties of logarithms to break it down further.

First, you can use the quotient rule, which states that log base b (a/c) is equal to log base b (a) - log base b (c). Applying this rule to the expression log base 10 (9/300), we get:

log base 10 (9/300) = log base 10 (9) - log base 10 (300)

Next, you can use the power rule, which states that log base b (a^m) is equal to m * log base b (a). Applying this rule to the first term, log base 10 (9), we have:

log base 10 (9) = 2 * log base 10 (3) (assuming you meant log base 10 (9) = 2 * log base 10 (3))

Lastly, you can use the addition rule, which states that log base b (a) + log base b (c) is equal to log base b (a * c). Applying this rule to the second term, log base 10 (300), we have:

log base 10 (300) = log base 10 (3 * 100) = log base 10 (3) + log base 10 (100) = log base 10 (3) + 2

Now, putting the simplified terms together, we have:

log base 10 (9/300) = 2 * log base 10 (3) - (log base 10 (3) + 2)

Combining like terms, we get:

= 2 * log base 10 (3) - log base 10 (3) - 2

Simplifying further:

= log base 10 (3) - 2

Therefore, the simplified form of the logarithmic expression log base 10 (9/300) is log base 10 (3) - 2.