diff --git a/r/R/dplyr-functions.R b/r/R/dplyr-functions.R
index d2f7892aee8..808956efe15 100644
--- a/r/R/dplyr-functions.R
+++ b/r/R/dplyr-functions.R
@@ -694,6 +694,23 @@ nse_funcs$wday <- function(x, label = FALSE, abbr = TRUE, week_start = getOption
 }
 
 nse_funcs$log <- nse_funcs$logb <- function(x, base = exp(1)) {
+  # like other binary functions, either `x` or `base` can be Expression or double(1)
+  if (is.numeric(x) && length(x) == 1) {
+    x <- Expression$scalar(x)
+  } else if (!inherits(x, "Expression")) {
+    arrow_not_supported("x must be a column or a length-1 numeric; other values")
+  }
+
+  # handle `base` differently because we use the simpler ln, log2, and log10
+  # functions for specific scalar base values
+  if (inherits(base, "Expression")) {
+    return(Expression$create("logb_checked", x, base))
+  }
+
+  if (!is.numeric(base) || length(base) != 1) {
+    arrow_not_supported("base must be a column or a length-1 numeric; other values")
+  }
+
   if (base == exp(1)) {
     return(Expression$create("ln_checked", x))
   }
@@ -705,8 +722,8 @@ nse_funcs$log <- nse_funcs$logb <- function(x, base = exp(1)) {
   if (base == 10) {
     return(Expression$create("log10_checked", x))
   }
-  # ARROW-13345
-  arrow_not_supported("`base` values other than exp(1), 2 and 10")
+
+  Expression$create("logb_checked", x, Expression$scalar(base))
 }
 
 nse_funcs$if_else <- function(condition, true, false, missing = NULL) {
diff --git a/r/tests/testthat/test-dplyr.R b/r/tests/testthat/test-dplyr.R
index d3a9994b5f1..ab7296b7818 100644
--- a/r/tests/testthat/test-dplyr.R
+++ b/r/tests/testthat/test-dplyr.R
@@ -1035,12 +1035,60 @@ test_that("log functions", {
     df
   )
 
+  # test log(, base = (length == 1))
+  expect_dplyr_equal(
+    input %>%
+      mutate(y = log(x, base = 5)) %>%
+      collect(),
+    df
+  )
+
+  # test log(, base = (length != 1))
   expect_error(
-    nse_funcs$log(Expression$scalar(x), base = 5),
-    "`base` values other than exp(1), 2 and 10 not supported by Arrow",
+    nse_funcs$log(10, base = 5:6),
+    "base must be a column or a length-1 numeric; other values not supported by Arrow",
     fixed = TRUE
   )
 
+  # test log(x = (length != 1))
+  expect_error(
+    nse_funcs$log(10:11),
+    "x must be a column or a length-1 numeric; other values not supported by Arrow",
+    fixed = TRUE
+  )
+
+  # test log(, base = Expression)
+  expect_dplyr_equal(
+    input %>%
+      # test cases where base = 1 below
+      filter(x != 1) %>%
+      mutate(
+        y = log(x, base = x),
+        z = log(2, base = x)
+      ) %>%
+      collect(),
+    df
+  )
+
+  # log(1, base = 1) is NaN in both R and Arrow
+  # suppress the R warning because R warns but Arrow does not
+  suppressWarnings(
+    expect_dplyr_equal(
+      input %>%
+        mutate(y = log(x, base = y)) %>%
+        collect(),
+      tibble(x = 1, y = 1)
+    )
+  )
+
+  # log(n != 1, base = 1) is Inf in R and Arrow
+  expect_dplyr_equal(
+    input %>%
+      mutate(y = log(x, base = y)) %>%
+      collect(),
+    tibble(x = 10, y = 1)
+  )
+
   expect_dplyr_equal(
     input %>%
       mutate(y = logb(x)) %>%