# How do you use log10^5=.6990 and log10^7=.8451 to evaluate the expression log_10 35?

May 3, 2016

Log form and index form are interchangeable.
Remember that logs are indices - the same laws apply.

#### Explanation:

If ${a}^{b} = c \Rightarrow {\log}_{a} c = b$

Let's change from log from to index form first.

${\log}_{10} 5 = 0.6990 \Rightarrow {10}^{0.6990} = 5$
${\log}_{10} 7 = 0.8451 \Rightarrow {10}^{0.8451} = 7$

$35 = 5 \times 7 = {10}^{0.6990} \times {10}^{0.8451}$

If the bases (10) are the same and you are multiplying, add the indices.

${10}^{1.5441} = 35$ This is index form - change to log form:

${\log}_{10} 35 = 1.5441$